show that (Cos 7theta- iSin 7theta)² (Cos 3theta+ iSin 3theta)⁴ /(cos 3theta+ iSin 5theta)³ (Cos theta- iSin theta)^7 =1
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Solve:
cosθ+cos7θ+cos3θ+cos5θ=0
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Answer
We have,
cosθ=cos7θ=cos3θ+cos5θ=0
We know that,
cosC+cosD=2cos
2
C+D
cos
2
C−D
Now,
cosθ+cos7θ+cos3θ+cos5θ=0
2cos
2
θ+7θ
cos
2
θ−7θ
+2cos
2
3θ+5θ
cos
2
3θ−5θ
=0
2cos4θcos(−3θ)+2cos4θcos(−θ)=0
2cos4θ[cos3θ+cosθ]=0
∴cos(−θ)=cosθ
Again,
2cos4θ[2cos
2
3θ+θ
cos
2
3θ−θ
]=0
2cos4θ[2cos2θcosθ]=0
4cos4θcos2θcosθ=0
cos4θcos2θcosθ=0
If cos4θ=0
cos4θ=cos90
o
cos4θ=cos
2
π
4θ=
2
π
θ=
8
π
If cos2θ=0
cos2θ=cos
2
π
2θ=
2
π
θ=
4
π
If cosθ=0
cosθ=cos
2
π
θ=
2
π
Hence, θ=
2
π
,
4
π
,
8
π
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