How to prove $\left| a_v\cos{vx}+b_v\sin{vx} \right|^2\leq a_v^2+b_v^2$

Question

My question is from the proof of corollaries of the main theorem on fourier series. How to prove $$\left| a_v\cos{vx}+b_v\sin{vx} \right|^2\leq a_v^2+b_v^2$$

Answer 1

$$(a\cos x + b\sin x)^2+(b\cos x - a\sin x)^2=a^2+b^2\implies(a\cos x + b\sin x)^2\le a^2+b^2$$