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

Question

Prove the following identities

\(\cfrac{(1+cot\text x+tan\,\text x)(sin\,\text x-cos\,\text x)}{sec^3\,\text x-cosec^3\,\text x}\) = sin²x cos²x

Answer 1

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

We know that

cosec θ = \(\cfrac1{sin\theta}\); sec θ = \(\cfrac1{cos\theta}\); 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)

= sin²x cos²x

= RHS

Hence proved.