Prove the identity: tan x/(1 - cot x) + cot x/(1 - tan x) = (sec x cosec x + 1)

Question

Prove the identity: \({\frac{tan x}{1 - cot x}} + {\frac{cot x}{1 - tan x}}\)= (sec x cosec x + 1)

Answer 1

Let us consider LHS:

\({\frac{tan x}{1 - cot x}} + {\frac{cot x}{1 - tan x}}\)

On using the formula,

tan θ = sin θ/cos θ;

cot θ = cos θ/sin θ

Then,

On using the formula, a³ – b³ = (a – b) (a² + b² + ab)

On using the formula,

cosec θ = 1/sin θ,

sec θ = 1/cos θ;

cosec x × sec x + 1

sec x cosec x + 1

= RHS

∴ LHS = RHS

Thus proved.