Prove the identity sin5 theta/sin theta -cos 5 theta/cos theta is equal to 4 cos 2 theta
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Given : Sin5θ/Sinθ - Cos5θ/Cosθ = 4Cos2θ
To Find : Prove the identity
Solution:
Sin5θ/Sinθ - Cos5θ/Cosθ = 4Cos2θ
LHS =
Sin5θ/Sinθ - Cos5θ/Cosθ
= (Sin5θCosθ - Cos5θSinθ)/SinθCosθ
Using Sin(A - B) = SinACosB - CosASinB
= Sin(5θ -θ)/SinθCosθ
= Sin4θ /SinθCosθ
Using Sin2x = 2SinxCosx
= 2Sin2θCos2θ/SinθCosθ
= 2(2SinθCosθ).Cos2θ/SinθCosθ
= 2(2)Cos2θ
= 4Cos2θ
= RHS
QED
Hence Proved
Sin5θ/Sinθ - Cos5θ/Cosθ = 4Cos2θ
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Answer:
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