Chapter 6 - Analytic Trigonometry - Section 6.4 Trigonometric Identities - 6.4 Assess Your Understanding - Page 498: 91

See below.

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