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Question

Prove the following identities (1-17)

cos θ1-sin θ=1+cos θ+sin θ1+cos θ-sin θ

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Solution

RHS=1+cos θ+sin θ1+cos θ-sin θ =1+cos θ+sin θ1+cos θ-sin θ =1+cos θ+sin θ1+cos θ+sin θ1+cos θ-sin θ1+cos θ+sin θ =1+cos θ+sin θ21+cos θ2-sin θ2 =1+cos θ2+sin θ2+21+cos θsin θ12+cos2 θ+2×1×cos θ-sin2 θ =1+cos2 θ+2cos θ+sin2 θ+2sin θ cos θ+2sin θ1+cos2 θ+2cos θ-sin2 θ =1+sin2 θ+cos2 θ+2cos θ+2sin θ cos θ+2sin θ1-sin2 θ+cos2 θ+2cos θ =1+1+2cos θ+2sin θ cos θ+2sin θcos2 θ+cos2 θ+2cos θ =2+2cos θ+2sin θ cos θ+2sin θ2cos2 θ+2cos θ =1+cos θ+sin θ cos θ+sin θcos2 θ+cos θ =11+cos θ+sin θ cos θ+1cos θcos θ+1 =cos θ+11+sin θcos θcos θ+1 =1+sin θcos θ =1+sin θ×cos θcos θ×cos θ =1+sin θ×cos θcos2 θ =1+sin θ×cos θ1-sin2 θ =1+sin θ×cos θ1+sin θ1-sin θ =cos θ1-sin θ =LHSHence proved.

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