Chapter 6 - Analytic Trigonometry - Section 6.4 Trigonometric Identities - 6.4 Assess Your Understanding - Page 498: 92

See below.

$LHS=(2a\ sin\theta cos\theta)^2+a^2(cos^2\theta -sin^2\theta)^2 =4a^2sin^2\theta cos^2\theta+a^2cos^4\theta+a^2sin^4\theta-2a^2sin^2\theta cos^2\theta =a^2cos^4\theta+a^2sin^4\theta+2a^2sin^2\theta cos^2\theta =a^2(cos^2\theta+sin^2\theta)^2=a^2=RHS$