Prove the following identities ((tan x)/(1 - cot x)) + ((cot x)/(1 - tanx)) = (sec x cosec x + 1)

Question

Prove the following identities

\(\cfrac{tan\,\text x}{1-cot\,\text x}+\cfrac{cot\text x}{1-tan\,\text x}\) = (sec x cosec x + 1)

Answer 1

LHS = \(\cfrac{tan\,\text x}{1-cot\,\text x}+\cfrac{cot\text x}{1-tan\,\text x}\)

We know that tan θ = \(\cfrac{sin \,\theta}{cos\,\theta}\) and cot θ = \(\cfrac{cos\,\theta}{sin\,\theta}\)

We know that a³ - b³ = (a - b) (a² + b² + ab)

We know that sin²x + cos²x = 1.

We know that cosec θ = \(\cfrac1{sin \,\theta}\); sec θ = \(\cfrac1{cos \,\theta}\) 

= cosecx × secx + 1 

 secx cosecx + 1

= RHS

Hence proved.