Prove the identity: ((1 - sin x cos x)/(cos x(sec x - cosec x)).((sin^2x - cos^2 x)/(sin^3 x+ cos^3 x)) = sin x = sin x

Question

Prove the identity: \({\frac{1 - sin x cos x}{cosx(sec x - cosec x)}}\).\({\frac{sin^2 x - cos^2 x}{sin^3 x + cos^3 x}}\) = sin x

Answer 1

Let us consider LHS:

 \({\frac{1 - sin x cos x}{cosx(sec x - cosec x)}}\).\({\frac{sin^2 x - cos^2 x}{sin^3 x + cos^3 x}}\) 

On using the formula,

cosec θ = 1/sin θ;

sec θ = 1/cos θ;

Then,

sin x

= RHS

∴ LHS = RHS

Thus proved.