The sum of all values of
Question:

The sum of all values of $\theta \in\left(0, \frac{\pi}{2}\right)$ satisfying

$\sin ^{2} 2 \theta+\cos ^{4} 2 \theta=\frac{3}{4}$ is :

  1. $\frac{\pi}{2}$

  2. $\pi$

  3. $\frac{3 \pi}{8}$

  4. $\frac{5 \pi}{4}$


Correct Option: 1

Solution:

$\sin ^{2} 2 \theta+\cos ^{4} 2 \theta=\frac{3}{4}, \theta \in\left(0, \frac{\pi}{2}\right)$

$\Rightarrow 1-\cos ^{2} 2 \theta+\cos ^{4} 2 \theta=\frac{3}{4}$

$\Rightarrow 4 \cos ^{4} 2 \theta-4 \cos ^{2} 2 \theta+1=0$

$\Rightarrow\left(2 \cos ^{2} 2 \theta-1\right)^{2}=0$

$\Rightarrow \cos ^{2} 2 \theta=\frac{1}{2}=\cos ^{2} \frac{\pi}{4}$

$\Rightarrow \quad 2 \theta=\mathrm{n} \pi \pm \frac{\pi}{4}, \mathrm{n} \in \mathrm{I}$

$\Rightarrow \quad \theta=\frac{\mathrm{n} \pi}{2} \pm \frac{\pi}{8}$

$\Rightarrow \quad \theta=\frac{\pi}{8}, \frac{\pi}{2}-\frac{\pi}{8}$

Sum of solutions $\frac{\pi}{2}$

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