Number of distinct solutions of equation frac{5}{4} cos ^2 2 x+cos ^4 x+sin ^4 x+cos ^6 x+sin ^6 x=2

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The number of distinct solutions of the equation $\frac{5}{4} \cos ^2 2 x+\cos ^4 x+\sin ^4 x+\cos ^6 x+\sin ^6 x=2$ in the interval $[0,2 \pi]$ is

A

$5$

B

$6$

C

$7$

D

$8$

(IIT-2015)

Solution

$\frac{5}{4} \cos ^2 2 x+\cos ^4 x+\sin ^4 x+\cos ^6 x+\sin ^6 x=2$

$\Rightarrow \frac{5}{4} \cos ^2 2 x-5 \cos ^2 x \sin ^2 x=0$

$\Rightarrow \tan ^2 2 x=1, \text { where } 2 x \in[0,4 \pi]$

$\text { Number of solutions }=8$

Standard 11

Mathematics

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