Prove the identity whether is true or false cos⁡2θ=1−tan2θ1+tan2θ

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Prove the identity whether is true or false  cos⁡2θ=1−tan2θ1+tan2θ

Jozlyn · Accepted Answer

Given that:  The trigonometry identity cos⁡2θ=1−tan2θ1+tan2θ  Use:  The trigonometry formula:  tan⁡θ=sin⁡θcos⁡θ  sin2θ+cos2θ=1  cos⁡2θ=cos2θ−sin2θ  To prove whether the given identity is true or false.  Consider the right hand side of the identity.  RHS =1−tan2θ1+tan2θ  =1−(sin⁡θcos⁡θ)21+(sin⁡θcos⁡θ)2  =1−sin2θcos2θ1+sin2θcos2θ  =cos2θ−sin2θcos2θcos2θ+sin2θcos2θ  =(cos2θ−sin2θ)cos2θcos2θ(cos2θ+sin2θ)  =cos2θ−sin2θcos2θ+sin2θ  Use the above trigonometry formula:  cos2θ−sin2θ=cos⁡2θ and cos2θ+sin2θ=1  To get,  RHS =cos⁡2θ1  =cos⁡2θ  To get,  RHS =cos⁡2θ  To get,  LHS = RHS  Thus,  The given identity is true and it is true for all 0.  Therefore,  cos⁡2θ=1−tan2θ1+tan2θ is true of all 0.

Prove the identity whether is true or false. \cos2\theta=\frac{1-\tan^2\theta}{1+\tan^2\theta}

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