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Question

If un=sinnθ+cosnθ,then show that u5u7u3u5=u3u1.

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Solution

un=sinnθ+cosnθ
Now u5u7u3u5
=sin5θ+cos5θsin7θcos7θcos3θ+sin3θsin5θcos5θ
=sin5θ+cos5θsin5θ(1cos2θ)cos5θ(1sin2θ)cos3θ+sin3θsin3θ(1cos2θ)cos3θ(1sin2θ)
=sin5θcos2θ+cos5θsin2θsin3θcos2θ+cos3θsin3θ
=(sin3θ+cos3θ)(sin2θcos2θ)(sinθ+cosθ)(sin2θcos2θ)=u3u1

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