1. Tardigrade
  2. Question
  3. Mathematics
  4. Let S= θ ∈[-π, π]- ± (π/2): sin θ tan θ+ tan θ= sin 2 θ . If T = displaystyle∑θ ∈ S cos 2 θ, then T + n ( S ) is equal

Question Error Report

Thank you for reporting, we will resolve it shortly

Back to Question

Q. Let $S=\left\{\theta \in[-\pi, \pi]-\left\{\pm \frac{\pi}{2}\right\}: \sin \theta \tan \theta+\tan \theta=\sin 2 \theta\right\} .$ If $T =\displaystyle\sum_{\theta \in S } \cos 2 \theta$, then $T + n ( S )$ is equal

JEE MainJEE Main 2022Trigonometric Functions

Solution:

$\sin \theta \tan \theta+\tan \theta=\sin 2 \theta$
$\tan \theta(\sin \theta+1)=\frac{2 \tan \theta}{1+\tan ^{2} \theta}$
$\tan \theta=0 \Rightarrow \theta=-\pi, 0, \pi$
$(\sin \theta+1)=2 \cdot \cos ^{2} \theta=2(1+\sin \theta)(1-\sin \theta)$
$\sin \theta=-1$ which is not possible
$\sin \theta=\frac{1}{2} \theta=\frac{\pi}{6}, \frac{5 \pi}{6}$
$n ( s )=5$
$T =\cos 0+\cos 2 \pi+\cos 2 \pi+\cos \frac{\pi}{3}+\cos \frac{5 \pi}{3}$
$T =4$
$T = n ( s )=9$