Let S=left[-π, frac{π}{2}right)-left{-frac{π}{2},-frac{π}{4},-frac{3 π}{4}, frac{π}{4}right} .

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Trigonometrical Equations

hard

Let $S=\left[-\pi, \frac{\pi}{2}\right)-\left\{-\frac{\pi}{2},-\frac{\pi}{4},-\frac{3 \pi}{4}, \frac{\pi}{4}\right\}$. Then the number of elements in the set $=\{\theta \in S : \tan \theta(1+\sqrt{5} \tan (2 \theta))=\sqrt{5}-\tan (2 \theta)\}$ is $...$

A

$0$

B

$5$

C

$3$

D

$4$

(JEE MAIN-2022)

Solution

$\tan \theta+\sqrt{5} \tan 2 \theta \tan \theta=\sqrt{5}-\tan 2 \theta$ $\tan 3 \theta=\sqrt{5}$

$\theta=\frac{ n \pi}{3}+\frac{\alpha}{3} ; \quad \tan \alpha=\sqrt{5}$

Five solution

Standard 11

Mathematics

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(JEE MAIN-2022)