Question

Prove that:
(A cos x + B sin x)^2 - (B cos x + A sin x)^2 = (A^2 - B^2) (cos^2x - sin^2x)

Answer 1

see explanation

Explanation:

Consider the left side

(Acosx + Bsinx)² - (Bcosx + Asinx)² ← expand factors using FOIL

= A²cos²x + 2ABsinxcosx + B²sin²x - (B²cos²x + 2ABsinxcosx + A²sin²x)

= A²cos²x + 2ABsinxcosx + B²sin²x - B²cos²x - 2ABsinxcosx - A²sin²x

= A²cos²x - Bcos²x + B²sin²x - A²sin²x ← factor first/second and third/fourth terms

= cos²x(A² - B²) - sin²x(A² - B²) ← factor out (A² - B²) from each term

= (A² - B²)(cos²x - sin²x)

= right side ⇒ proven