Embibe Experts Solutions for Chapter: Trigonometric Equations and Inequalities, Exercise 1: JEE Main - 24 June 2022 Shift 1

Let $S=\left\{\theta \in \left[0,2\pi \right]:8^{2{\sin }^2\theta }+8^{2{\cos }^2\theta }=16\right\}$. Then $n\left(S\right)+\sum _{\theta \in \mathrm{S}}\left(\sec \left(\frac{\pi }{4}+2\theta \right)cosec\left(\frac{\pi }{4}+2\theta \right)\right)$ is equal to:

If the sum of solutions of the system of equations $2{\sin }^2\theta -\cos 2\theta =0$ and $2{\cos }^2\theta +3\sin \theta =0$ in the interval $\left[0,2\pi \right]$ is $k\pi$, then $k$ is equal to _______.

Let $S=\left\{\theta \in \left(0,\frac{\pi }{2}\right):\sum _{m=1}^9\sec \left(\theta +\left(m-1\right)\frac{\pi }{6}\right)\sec \left(\theta +\frac{m\pi }{6}\right)=-\frac{8}{\sqrt{3}}\right\}$. Then

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 $A=\left\{\theta \in \mathrm{S}:\tan \theta \left(1+\sqrt{5}\tan \left(2\theta \right)\right)=\sqrt{5}-\tan \left(2\theta \right)\right\}$ is _____ .

Let $S=\left\{\theta \in (0,2\pi ):7{\cos }^2\theta -3{\sin }^2\theta -2{\cos }^{2}2\theta =2\right\}$. Then, the sum of roots of all the equations $x^2-2\left({\tan }^2\theta +{\cot }^2\theta \right)x+6{\sin }^2\theta =0 \theta \in \mathrm{S}$, is _______.

Let $S=\left\{\theta \in [0,2\mathrm{\pi }): \tan \left(\mathrm{\pi cos\theta }\right)+\tan \left(\mathrm{\pi sin\theta }\right)=0\right\}$, then $\sum _{\theta \in S}{\sin }^2\left(\theta +\frac{\mathrm{\pi }}{4}\right)$ is equal to

Let $f\left(\theta \right)=3\left({\sin }^4\left(\frac{3\pi }{2}-\theta \right)+{\sin }^4(3\pi +\theta )\right)-2\left(1-{\sin }^{2}2\theta \right)$ and $S=\left\{\theta \in [0,\pi ]:f^{\text{'}}\left(\theta \right)=-\frac{\sqrt{3}}{2}\right\}$.

If $4\beta =\sum _{\theta \in S}\theta$  then $f\left(\beta \right)$ is equal to

If the solution of the equation ${\log }_{\cos x}\left(\cot x\right)+4{\log }_{\sin x}\left(\tan x\right)=1, x\in \left(0,\frac{\pi }{2}\right)$ is ${\sin }^{-1}\left(\frac{\alpha +\sqrt{\beta }}{2}\right)$, where $\alpha ,\beta$ are integers, then $\alpha +\beta$ is equal to: