The sum of all values of (0,2) satisfying ^22+^42=34 is:

Question

The sum of all values of $	heta\in\left(0,\dfrac{\pi}{2}ight)$ satisfying $\sin^22	heta+\cos^42	heta=\dfrac34$ is:

Accepted Answer

$1-\cos^22\theta+\cos^42\theta = \tfrac34 \Rightarrow \cos^22\theta(1-\cos^22\theta)=\tfrac14$.

By AM--GM, $\cos^22\theta(1-\cos^22\theta) \le \tfrac14$ always, with equality iff $\cos^22\theta=\tfrac12$, giving $2\theta=\pi/4$ or $3\pi/4$, i.e. $\theta=\pi/8, 3\pi/8$.

$$\text{Sum} = \frac{\pi}{8}+\frac{3\pi}{8} = \frac{\pi}{2}.$$

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