Prove the following identities: sin^(π/8 + x/2) - sin^2(π/8 - x/2) = 1/√2 sin x

Question

Prove the following identities:

\(sin^2 \left(\frac{\pi}{8}+\frac{x}{2}\right)-sin^2 \left(\frac{\pi}{8}-\frac{x}{2}\right) = \frac{1}{\sqrt{2}}sin\,x\)

Answer 1

To prove: \(sin^2 \left(\frac{\pi}{8}+\frac{x}{2}\right)-sin^2 \left(\frac{\pi}{8}-\frac{x}{2}\right) = \frac{1}{\sqrt{2}}sin\,x\)

Proof:

Take LHS:

\(sin^2 \left(\frac{\pi}{8}+\frac{x}{2}\right)-sin^2 \left(\frac{\pi}{8}-\frac{x}{2}\right)\)

Identities used:

sin²A - sin²B = sin (A + B) sin(A – B)

Therefore,