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The sum of all values of $\theta\epsilon (0,\pi/2)$ satisfying $sin^{2}2\theta +cos^{4}2\theta = 3/4$ is:
Option 1)

$\pi$
Option 2)

$5\pi/4$
Option 3)

$\pi/2$
Option 4)

$3\pi/8$

Answers (1)

best_answer

Trigonometric Identities -

$\sin ^{2}\Theta + \cos ^{2}\Theta = 1$

$1 + \tan ^{2}\Theta = \sec ^{2}\Theta$

$1 + \cot ^{2}\Theta = cosec ^{2}\Theta$

- wherein

They are true for all real values of $\Theta$

Graph of Trigonometric Ratios -

Graph 2

- wherein

This is the graph of $y = \cos x$

??????Given that

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

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

$\because sin^{2}\theta + cos^{2}\theta = 1$

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

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

$cos^{2}(2\theta) = \frac{1}{2} = cos^{2}\left ( \frac{\pi}{4} \right )$

$2\theta= n\pi \pm \frac{\pi}{4} , n\epsilon I$

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

as , $\theta \epsilon \left ( 0, \frac{\pi}{2} \right )$

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

Sum = $\frac{\pi}{2}$

Option 1)

$\pi$

Option 2)

$5\pi/4$

Option 3)

$\pi/2$
Option 4)

$3\pi/8$

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