Prove that 1+2sin⁡θcos⁡θcos2θ−sin2θ=1+tan⁡θ1−tan⁡θ

Gerald Ritter

Gerald Ritter

Answered

2022-01-28

Prove that 1+2sinθcosθcos2θsin2θ=1+tanθ1tanθ

Answer & Explanation

Howard Gallagher

Howard Gallagher

Expert

2022-01-29Added 13 answers

Note that
1+2sinθcosθ=cos2θ+sin2θ+2sinθcosθ=(cosθ+sinθ)2
Therefore
1+2sinθcosθcos2θsin2θ=(cosθ+sinθ)2(cosθ+sinθ)(cosθsinθ)
=cosθ+sinθcosθsinθ
=cosθ(1+tanθ)cosθ(1tanθ)
=1+tanθ1tanθ

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