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

Question

Prove the following identities 

\(\cfrac{1-sin\,\text x\,cos\,\text x}{cos\,\text x(sec\,\text x-cosec\,\text x)}\).\(\cfrac{sin^2\text x-cos^2\text x}{sin^3\text x+cos^3\text x}=sin\,\text x\)

Answer 1

LHS = \(\cfrac{1-sin\,\text x\,cos\,\text x}{cos\,\text x(sec\,\text x-cosec\,\text x)}\) x \(\cfrac{sin^2\text x-cos^2\text x}{sin^3\text x+cos^3\text x}\)

 We know that

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

We know that sin²x + cos²x = 1.

= sinx

= RHS

Hence proved.