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

Question

Prove the following identities

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

Answer 1

LHS = 

Prove the following identities

\(\cfrac{sin^3\text x+cos^3\text x}{sin\text x+cos\,\text x}\) +  \(\cfrac{sin^3\text x-cos^3\text x}{sin\text x-cos\,\text x}\) 

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

We know that sin²x + cos²x = 1.

= 1 - sinx cosx + 1 + sinx cosx

= 2

= RHS

Hence proved.