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Question
Prove that following identities:
sin 5θ=5 sin θ−20 sin3θ+16 sin5θ
Prove that following identities:
sin 5θ=5 sin θ−20 sin3θ+16 sin5θ
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Solution
L.H.S., sin 5θ=sin(3θ+2θ)
=sin 3θ cos 2θ+cos 3θ. sin 2θ=(3 sin θ−4 sin3θ)(1−2 sin2θ)(4cos3θ−3 cos θ)2 sin θcos θ=3 sin θ−4 sin3θ−6 sin3θ+8 sin5θ+(8 cos4θ−6 cos2 θ)sin θ=3 sin θ−10 sin3θ+8sin5θ+8 sin θ[(1−sin2θ)2−6 sin θ(1−sin2θ)]=3 sin θ−10 sin3 θ+8 sin5 θ+8 sin θ−16 sin3 θ+8 sin5θ−6 sin θ+6 sin3θ=5 sin θ−20 sin3θ+16 sin5 θ=RHS
L.H.S., sin 5θ=sin(3θ+2θ)
=sin 3θ cos 2θ+cos 3θ. sin 2θ=(3 sin θ−4 sin3θ)(1−2 sin2θ)(4cos3θ−3 cos θ)2 sin θcos θ=3 sin θ−4 sin3θ−6 sin3θ+8 sin5θ+(8 cos4θ−6 cos2 θ)sin θ=3 sin θ−10 sin3θ+8sin5θ+8 sin θ[(1−sin2θ)2−6 sin θ(1−sin2θ)]=3 sin θ−10 sin3 θ+8 sin5 θ+8 sin θ−16 sin3 θ+8 sin5θ−6 sin θ+6 sin3θ=5 sin θ−20 sin3θ+16 sin5 θ=RHS
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