Trigonometry (B.Sc. 1st year)
262 videos • 1,896 views • by Doctor of Mathematics
1
Prove that: log[1/(1-e^iθ)]=log(1/2cocθ)+i(π/2-θ/2)
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2
log[tan(π/4+x/2)]=i.tan-(sinhx) Prove that
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3
i^i^i......ad.inf.=A+iB ,principal values only being considered, prove that tan(pi.A/2)=B/A and A^2+
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4
Separate into real and imaginary parts of log[cos(x+iy)]
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5
If cos(x+iy)=cosA+isinA,prove that sin^4x=sin^2A
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6
If x=2cosA.coshB;y=2sinA.sinhB prove that (i)sec(A+iB)+sec(A-iB)=4/(x^2+y^2);(ii)sec(A+iB)-sec(A-iB)
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7
Find the all values of Log[exp(x+iy)] ;where x,y are real.
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8
Tanh(x+iy);Separate real and imaginary parts of tanh(x+iy).
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9
If tany=tanα.tanβ and tanz=cotα .tanhβ prove that tan(y+z)=sinh2β.cosec2α.
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10
If cos(A+iB)=cosα+isinα, show that (i) ±sin^2(A)=sinα (ii) ±sinh^2B=sinα (iii)cos2A+Cosh2B=2
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11
Resolve e^sinh(x+iy) into real and imaginary parts
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12
If cosh(u+iv)=x+iy,prove that (i)x^2/cosh^2u+y^2/sinh^2u (ii)x^2/cos^2v-y^2/sin^2v=1
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13
If u+iv=cos(x+iy),prove that (1+u)^2+v^2=(coshy+cosx)^2
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14
If cos-1(x+iy)=A+iB, show that x^2.secc^2A-y^2.cosec^2=1 and x^2.sech^2B+y^2.cosech^2B=1
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15
Prove that tanh(α+iβ)=sinh2α/(cosh2α+cos2β) +i.sin2β/(cosh2α+cos2β)
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16
If tan−1(e^ia)=θ+iϕ and α/=π/2,prove that(i)2ϕ =logtan(π/4+α/2)(ii)θ=nπ/2+π/4
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17
Separate into real and imaginary part of the quantity e^(x+iy)
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18
If A+iB=ρ.tan(x+iy),show that tanh2y=2ρ.B/(ρ²+A²+B²)
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19
Find the value of e^(±iπ/2) ||Best Mathematics channel on YouTube▶️||
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20
If tanx/2=tanhx/2,prove that cosx.coshx=1
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21
cosh2x+cosh5x+cosh8x+cosh11x=4cosh13x/2.cosh3x/2.cosh3x.
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22
If sin(x+iy)=p+iq,where p and q are real ,then find q.
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23
e^e^(x+iy)=e^e^x.cosy.[cos(e^x.siny)+isin(e^x.siny)]
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24
exp(-π/2)=-i=exp(i.3π/2)
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25
exp(2npi)=1 B.Sc. Trigonometry
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26
Separate real and imaginary parts cot(a-ib)
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27
exp[cos(x+iy)] separate real and imaginary parts
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28
exp[(a+ib)(x+iy)], separate into real and imaginary parts
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29
{sin(α-θ)+e^(±iα).sinθ}^n=sin^n-1(α).[sin(α-nθ)+sin(nθ).e^(±iα)]
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30
icos(α+iβ)=icosα.coshβ+sinα.sinhβ
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31
If tan(α+iβ)=i,prove that α is indeterminant and β is infinite.
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32
cos(α+iβ)+isin(α+iβ)=e^(-β)[cosα+isinα]
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33
If α and β are imaginary cube root unity ,prove that αe^αx+βe^βx=-e^(-x/2).[cos-/3x/2+-/3sin-/3x/2]
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34
cosh2x+sinh2x=(1+tanhx)/(1-tanhx)
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35
1/2(coshx+cosx)=1+x^4/4!+x^8/8!+.....
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36
Find the value of Log(1+i)
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37
If (a1+ib1)(a2+ib2)(a3+ib3)..(an+ibn)=A+iB,prove that tan-1b1/a1+tan-1b2/a2+...+tan-1bn/an=tan-1B/A
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38
Log i =(4n+1)πi/2
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39
log(1+itanα)=logsecα+ia and deduce the expression for α and logcosα in powers of tanα
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40
tan(α+iβ)=sin2α/(cos2α+cosh2β)+isinh2β/(cos2α+cosh2β)
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41
Log[(a-b)+i(a+b)/(a+b)+i(a-b)]=i[2nπ+tan-1(2ab/a^2-b^2)]
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42
If sin(log(x+iy))=u+iv,prove that u^2/sin^2θ-v^2/cos^2θ=1 where x^2+y^2=e^2θ.
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43
log[sin(x+iy)/sin(x-iy)]=2itan-1(cotx.tanhy) prove that
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44
logcos(x+iy)1/2.logcos2x+cosh2y/2-itan-1(tanx.tanhy) prove that
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45
If A+iB=log(x+iy),show that B=tan-1y/x andA=1/2.log(x^2+y^2)
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46
log(1+re^iθ)=1/2log(1+2rcosθ+r^2)+itan-1(rsinθ/1+rcosθ).
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47
log(1+cos2θ+isin2θ)=log(2cosθ)+i(θ+2k.pi)
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48
sin[ilog(1+sinθ-icosθ)/(1+sinθ+icosθ)]=cosθ
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49
Separate exp.{cos(x+iy)} into real and imaginary parts.
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50
exp{sinh(x+iy)} separate into real and imaginary parts.
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51
2 October 2020
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52
If tanlog(x+iy)=a+ib,prove that tan[log(x^2+y^2)]=2a/1-a^2-b^2
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53
Log(Log i)=log[(4n+1)pi/2]+(4m+1)/2.pi.i where n and m are any two integers
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54
If ρ(cosα+isinα)=sin(θ+iϕ),prove that ρ^2=1/2(cosh2ϕ-cos2θ) and tanα=cotθ.tanhϕ.
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55
If cos(x+iy)=A+iB,prove that A^2/cos^2(x)-B^2/sin^2(x)=1
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56
If z=x+iy , find the modulus of e^z^2
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57
If cosh(u+iv)=x+iy,prove thatx^2/cos^2v-y^2/sin^2v=1 and x^2/cosh^2+y^2/sinh^2u=1.
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58
Gregory's series Proof in Hindi/Urdu
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59
π=2√3[1-1/3^2+1/5.3^2-1/7.3^3+.....]
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60
tan-1x/x+tan-1y/y+tan-1z/z=3{1-1/7+1/13-1/19+1/25......}
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61
If tan(θ+iϕ)=tanα+i.secα,prove that e^2ϕ=±cotα/2 and 2θ=nπ+(π/2+α)
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62
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63
2(1/3.5+1/7.9+1/11.13+.....)=π/4
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64
π/12=(1-1/3^1/2)-1/3(1-1/3^3/2)+1/5(1-1/2^5/2)-........
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65
π/8=1/1.3+1/5.7+1/9.11+.....
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66
2[1/3-1/3.(1/3)^3+1/5(1/3)^5-....]+[1/7-1/3(1/7)^3+1/5(1/7)^5-....]=π/4
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67
tan-1x=π/4+(x-1/x+1)-1/3(x-1/x+1)^3+1/5(x-1/x+1)^5+......a Gregory's series problem
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68
If θ lies between 0 and π/2,prove that tan-1(1-cosθ/1+cosθ)=tan^2θ/2-1/3tan^6θ/2+1/5tan^10θ/2.......
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69
tan-1(cosθ+sinθ/cosθ-sinθ)=nπ+π/4+tanθ-1/3tan^3θ+1/5tan^5θ-.....
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70
θ^2=tan^2(θ)-(1+1/3)tan^4(θ)/2+(1+1/3+1/5)tan^6(θ)-... if θ greater than -π/4 and less than π/4
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71
ϕ=π/2-cotϕ+1/3cot^3(ϕ)-1/5cot^5ϕ+..... ,If ϕ lies between π/4 and 3π/4 show that
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72
π/2√3=1-1/3.3+1/5.3^2-1/7.3^3+......................
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73
For the validity of General Theorem θ-nπ=tanθ-1/3tan^3(θ)+1/5tan^5(θ)-......find value of n
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74
π/4=17/21-713/81×343+.....+(-1)^n+1/(2n+1){2/3.9^1-n+71-2n}
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75
1-1/4.4^2+1/5.4^4-1/7.4^6+......Find the sum of the series
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76
7/1.3.5+19/5.7.9.+31/9.11.13+.....Find the sum of series
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77
If x less than √2-1,prove that 2(x-x^3/3+x^5/5-.....)=2x/1-x^2-1/3(2x/1-x^2)^3+1/5(2x/1-x^2)^5-....
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78
1+1/3-15-1/7+...........=π/2√2 Prove that:B.Sc. Mathematics:Trigonometry:Gregory's series
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79
π/4=[1/2+1/5+1/8]-1/3[1/2^3+1/5^3+1/8^3]+1/5[1/2^5+1/5^5+1/8^5]+.....Show that:B.Sc. Mathematics:Tri
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80
If i^i^i......ad.inf.=A+iB
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81
Show that cosπ/7, cos3π/7 and cos5π/7 are the roots of the equation,8x^3-4x^2-4x+1=0 and deduce the
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82
Separate cos-1(cosθ+isinθ) into real and imaginary parts
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83
Prove that 1+cos10θ/1+cosθ=(16cos^4θ-20cos^2θ+5)^2
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84
Solve the equation x^9+1=0
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85
Expand cos^3θ in power of θ.
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86
sin^2θ.cosθ=θ^2-(5/6)θ^4+... +(-1)^n+1.3^2n-1/4.θ^2n/2n!+....
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87
Saap Ne Kya Bola? by Sayyed Aminul Qadri
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88
Expand sinθ/θ in powers of θ .
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89
Prove that the equation cos2θ+a.cosθ+b.sinθ+c=0 has the sum of the four roots is a multiple of 2π.
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90
(1-1/2^2)(1-1/3^2)(1-1/5^2)(1-1/7^2).......=6/π^2,where 2,3,5,7,.....are all prime numbers.
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91
show that tan-1(tanhβ.cotα)=tan-1(β/α)-sigma n=1to infinity tan-1(2αβ/n^2.π^2-α^2+β^2)
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92
If n is very large ,then prove that √1/2π(2n+1=[2.4.6........2n]/[1.3.5......(2n-1)]
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93
Prove that 1/1^2+1/2^2+1/3^2+......=π^2/6
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94
sin(α)+sin(α+β)+sin(α+2β)+sin(α+3β)+....upto n terms
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95
If sin(θ+iϕ)=r.cosα+i.r.sinα ,prove that 2r^2=cosh2ϕ-cos2θ and tanα=tanhϕ.tanθ
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96
√1+sin2α +√1+sin2(α+β) +√1+sin2(α+2β)+.........upto nterms. Find the sum
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97
Prove that 1-1/3^3+1/5^3+..........=π^3/32
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98
If cos(α+iβ)=e^(iθ),prove that sin^2(α)=±sinθ
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99
Separate cosech(x+iy) into real and imaginary parts
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100
Separate sech(x+iy) into real and imaginary parts
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101
If cos(θ+iϕ).cos (α+iβ)=1,prove that tanh^2(ϕ).cosh^2(β)=sin^2(α) and tanh^2(β).cosh^2(ϕ)=sin^2(θ).
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101
If cos(θ+iϕ).cos (α+iβ)=1,prove that tanh^2(ϕ).cosh^2(β)=sin^2(α) and tanh^2(β).cosh^2(ϕ)=sin^2(θ).
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102
State and prove De Moivre's Theorem
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103
Prove that tan−1n+cot−1 (n+1) = tan−1 (n 2 +n+1) and if tan−1 x+tan−1 y+tan−1 z = π/2
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104
Separate Cosh(x+iy) into real and imaginary parts.
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105
Separate sin(x+iy) into real and imaginary parts
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106
If tan(θ+iϕ)=cosα+isinα,prove that θ=nπ/2+π/4 and ϕ=1/2.logtan(π/4+α/2)
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107
If (1+i)^p+iq/(1-i)^p+iq=a+ib,prove that one value of tan-1(b/a) is pi.p/2+q.log2
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108
If coshu=secθ,show that u=logtan(π/4+θ/2) and tanh^2(u/2)=tan^2(θ/2)
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109
If sin(θ+iϕ)=cosα+isinα,prove thatcos^2(θ)=±sinα and sinh^2(ϕ)=±sinα
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110
logsin(x+iy)
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111
2sinα-3sin2α+4sin3α-. ...upto n terms
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112
cosπ/3+1/3cos2π/3+1/5cos3π/3+1/7cos4π/3+..........to infinity.(summation of series)
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113
If α =log[tan(π/4+θ/3)] prove rhat sinhα=tanθ
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114
Prove that tan-1(tanhy.cotx)=tan-1y/x-sigma n=1 to infinity tan-1[2xy/n^2pi^2-x^2+y^2]
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115
If u+iv=cot(x+iy); v=-sinh2y/(cosh2y-cos2x)
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116
Separate real and imaginary parts (1+c.cosα+ic.sinα)^1/2
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117
Separate into real and imaginarg parts e^iϕ/1-c.e^iϕ
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118
prove that cos[i.ln(a+ib)/(a-ib)]=(a^2-b^2)/(a^2+b^2)
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119
Prove that ln(1+r.e^iθ)=1/2ln(1+2r.cosθ+r^2)+i.tan-1(r.sinθ/1+r.cosθ)
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120
Prove that Sin-1(cosecθ)={2n+(-1)^n}π/2+i.(-1)^n.logcotθ/2
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121
Prove that tan-1(tan2θ+tanh2ϕ/tan2θ-tanh2ϕ)+tan-1(tanθ-tanhϕ/tanθ+tanϕ)=tan-1(cotθ.cothϕ)
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122
n.sinα+n(n+1)sin2α/1.2+n(n+1)(n+2)sin3α/1.2.3.+.......sum upto infinity
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123
If coshu=secθ then prove that sinhu=tanθ
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124
If sin(x+iy)=tanA+i.secA then prove that cos2x.cosh2y=3.
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125
If tan-1(tanx+i.secx)=a+ib, prove that a=(2n+1)π/4+x/2 and b=1/2logcotx/2
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126
Show that the principal value of (x+iy) ^(a+ib) is wholly real and imaginary according as 1/2b.log(
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127
I/log(1+i) =A+iB, find values of A and B.
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128
Explanation of Shab-e-miraj and Isra on basis on special theory of relativity
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129
tan-1(4/1+3.4) +tan-1(6/1+8.9)+tan-1(8/1+15.16) +... n terms
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130
Sn sum of n term sinθ+sin2θ+sin3θ+... prove that Lt n tends to infinity S1+S2+S3+.. +Sn/n=1/2.cotθ/2
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131
If tan-1(a+ib) =sin-1(x+iy), show that a^2+b^2=x^2+y^2/x^4+y^4+2x^2.y^2-2x^2+2y^2+1
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132
Prove that if (1+I.tanα) ^1+itanβ can have only real values, one of them is (secα) ^sec^2β
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133
If cos(α+iβ) =r(cosθ+isinθ), then find the β
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134
Sum the following series: sinhα+n.sinh2α+n(n-1)/2!. sinh3α+... to (n+1) terms, where n is a positive
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135
Separate exp{(x+iy)(α+iβ)} into real and imaginary psrts
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136
#bsctrigonometry Separate ecp(e^iθ) into real and imaginary parts
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137
Separate tanh-1z into real and imaginary parts.
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138
1+1/3.y.cosα+1.4/3.4.y^2.cos2α+1.3.7/3.6.9.y^3.cos3α+.....sum the series.
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139
Sinhα+nsinh2α+n(n-1)/2!sing3α+... to (n+1)terms, where n is a positive Integer. #summation_of_series
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140
Separate sinai/(x+iy) into real and Imaginary parts
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141
If logsin(θ+iΦ) =α+iβ ,prove cos(θ-β)=e^2Φ.cos(β+θ)
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142
tan-1(e^ix) -tan-1(e^-ix) =tan-1(i)
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143
Find the sum to n terms the series sin^3(x/3) +3sin^3(x/9) +9sin^3(x/27)+...
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144
Sum to n terms, and to infinity, the series 1+c.cos+c^2cos2+....where c is less than one numerically
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145
If tan(α+iβ) =tanθ+isecθ, show that e^2β=cotθ/2, α=nπ/2+π/4+θ/2
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146
Express tanh-1(x+iy) in form A+iB and hence the value of tanh-1(iy)
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147
tan-1[(tan2x+tanh2y) /(tan2x-tanh2y)] +tan-1[(tanx-tanhy) /(tanx+tanhy)] =tan-1(cotxcithy)
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148
If tan(θ+iΦ) =tanα+isecα, prove that 2θ=nπ+π/2+α
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149
cosα+cos(α+β)+cos(α+β)+..... upto n term. Summation of series
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150
sinx.sin2x+sin2x.sin3x+sin3x.sin4x+.... upto n-1
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151
loglogsin(x+iy) separate into real and imaginary parts
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152
If x+iy=ctan(u+iv), prove that x^2+y^2+c^2-2cxcoth2u=0
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153
logcos(x+iy) separate into real and imaginary parts
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154
tan-1[(tan2x+tanh2y) /(tan2x-tanh2y)] +tan-1[(tanx-tanhy) /(tanx+tanhy)] =tan-1(cotxcothy)
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155
If tan(π/4+iα) =re^iθ, show that (1) r=1,(2) tanθ=sinh2α, (3) tanhα=tanθ/2
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156
Solve :(1) sinhx=1 (2) coshx=1/√2 (3) tanhx=√3 (4) tanhx=0.75
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157
If sin(x+iy). sin(+i)=1, show that tanh^2y. cosh^2Φ=cos^2θ
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158
Prove that tan-1(1-cosθ/1+cosθ)=tan^2(θ/2)+tan^6(θ/2)+tan^10(θ/2)-....
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159
=-cot+1/3cot^-1/5cot^5-
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160
Prove that log(x+iy) =1/2log(x^2+y^2) +itan-1y/x
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161
If x+iy=cos(u+iv), prove that (i) (1+x)^2+y^2= (coshv+cosu)^2 (ii) (1-x)^2+y^2=(coshv-cosu)^2
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162
If tan(x+iy)=α+iβ, prove that α/sin2x=β/sinh2y=1/cos2x+cosh2y
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163
cosθ+cos3θ+cos5θ+........ upto n terms ,Sum to n terms (summation of series)
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164
Express log[log(cosθ+isinθ)]
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165
If x is a cute and y=log[tan(π/2+x/2)], prove that cosx. coshy=1
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166
π/12=(1-1/√3)-1/3(1-1/√3^3)+1/5(1-1/√3^5) -........
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167
1-1/3.4^2+1/5.4^2+...........=4.tan-1(1/4)
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168
1+xcosθ+x^2cos2θ+.. . . upto n terms (summation of series)
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169
sinα+csin(α+β)+c^2sin(α+2β) +.... upto n terms
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170
1/1.3+1/5.7+1/9/.11+..........
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171
1/2^3-1/3.2^7+1/5.2^11+........
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172
If θ is between nπ-π/4 and nπ+π/4, prov nπ+π/4+tanθ-1/3tan^3θ+1/5tan^5θ-=tan-1(cosθ+sinθ/cosθ-sinθ)
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173
Prove that ∑ from n=1 to n=m cos^nθ.cosnθ=cos^m+1θ.sinmθ/sinθ.||Top mathematics channel on YouTube
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174
If i^i^i... ad. inf. =A+iB, principal value only being considered, prove tanπA/2=B/A; A^2+B^2=e^-πB
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175
sinhx+sinh(x+y)+sinh(x+2y)+...... upto n terms
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176
coshx+cosh(x+y)+cosh(x+2y)+.......upto n terms
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177
If cos(θ+iΦ)=r(cosα+sinα) , prove that Φ=1/2logsin(θ-α)/(θ+α)
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178
Sum thee series: cosθ+cos3θ+cos5θ+.........
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179
tanh-1(tanθ/2)=1/2logtan(π/4+θ/2)
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180
3sinθ+5sin2θ+7sin3θ+......upto n terms(summation of series)
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181
If coshu=secθ, prove that u=logtan(π/4+θ/2)
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182
(i)sinx+sin3x+sin5x+........upto n terms,(ii)cosx+cos3x+cos5x+...upto n terms
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183
cosα.cos2α+cos2α.cos3α+cos3α.cos4α+..... up ton terms.
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184
Separate exp(siniθ) into real and imaginary parts
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185
sin^3α+sin^(α+β) +sin^3(α+2β) +....... upto n terms
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186
sin^2(α)+sin^2(2α)+sin^2(3α) +... upto n terms
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187
Express log{log(cosθ+sinθ)} in the form of A+iB.||Top Mathematics channel on YouTube
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188
Prove that: sinh-x=-i.sin-1(ix) ||Top mathematics channel on YouTube
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189
If u=logtan(π/4+x/2)=x+a3x^3+a5x^5+.prove that x=u-a3u^3+a5u^5-.. Top mathematics channel on YouTube
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190
Prove that tanh-1x=sinh-1x/√1-x^2
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191
cos+a.cos2θ/1! +a^2cos2θ/2! +.... sum the series||Best Mathematics channel on YouTube▶️ top
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192
If, sin(θ+iΦ)=cosα+isinα, prove that sinα=cosh^2θ=sinh^2Φ
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193
If logsin(θ+iΦ)=α+iβ,prove that (i) cos2θ=e^2Φ+e^2Φ-4e^2α(ii)Φ=1/2log[cos(θ-β)/cos(θ+β)]
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194
f(x)=x+a is a continuous function and take takes only rational values. If f(0)=3.Find value f(2).
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195
Show that 2i.tan-1{itan(π/4-θ)}=log tanθ
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196
Express tan-1(cosθ+isinθ) in the form A+iB,and deduce that(i)cosθ-1/3cos3θ+1/5cos5θ-..(ii)sinθ-1/3si
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197
Prove that sin-1(i)=2nπ-i.logical(√2-1)
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198
If tanhx=sinθ, show that sinhx=tanθ; coshx=seeθ
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199
If z=e^iα.Show that (z^2-1) /(z^2+1)=itanα
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200
If tan(A+iB)=x+iy, prove that (I) x/y=sin2A/sinh2B (ii) x/sin2A=y/sinh2B=1/cos2A+cosh2B
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