Prove that tan θ/(1 - cot θ) + cot θ/(1 - tan θ) = 1 + tan θ + cot θ

Answer:

[tan θ/(1 – cot θ) ]+[ cot θ/(1 – tan θ)] = [ tan θ/(1 – 1/tanθ) ] + [ (1/tan θ)/(1 – tan θ)]

= [tan2 θ/(tan θ-1)] + [1/tanθ(1-tan θ)]

= [tan2 θ/(tan θ-1)] – [1/tanθ(tan θ-1)]

= (tan3 θ- 1)/tanθ(tan θ-1)

= (tan θ – 1)(tan2 θ + tan θ + 1)/tan θ(tan θ – 1)

= (tan2 θ + tan θ + 1)/tan θ

= tan θ + 1 + 1/tanθ

= tan θ + 1 + cot θ

Hence proved.

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