Trigonometry · JEE Advanced previous year questions

The total number of real solutions of the equation $$\theta ={\tan }^{-1}(2\tan \theta )-\frac{1}{2}{\sin }^{-1}\left(\frac{6\tan \theta }{9+{\tan }^2\theta }\right)$$ is (Here, the inverse trigonometric functions ${\sin }^{-1}x$ and ${\tan }^{-1}x$ assume values in $\left[-\frac{\pi }{2},\frac{\pi }{2}\right]$ and $\left(-\frac{\pi }{2},\frac{\pi }{2}\right)$, respectively.)

1. A.1
2. B.2
3. C.3
4. D.5

Let $$\alpha =\frac{1}{\sin 60^{\circ }\sin 61^{\circ }}+\frac{1}{\sin 62^{\circ }\sin 63^{\circ }}+\cdots +\frac{1}{\sin 118^{\circ }\sin 119^{\circ }}.$$ Then the value of $${\left(\frac{\mathrm{cosec}{1}^{\circ }}{\alpha }\right)}^2$$ is ____.

Let $\frac{\pi }{2}<x<\pi$ be such that $\cot x=\frac{-5}{\sqrt{11}}$. Then $$\left(\sin \frac{11x}{2}\right)(\sin 6x-\cos 6x)+\left(\cos \frac{11x}{2}\right)(\sin 6x+\cos 6x)$$ is equal to

1. A.$\frac{\sqrt{11}-1}{2\sqrt{3}}$
2. B.$\frac{\sqrt{11}+1}{2\sqrt{3}}$
3. C.$\frac{\sqrt{11}+1}{3\sqrt{2}}$
4. D.$\frac{\sqrt{11}-1}{3\sqrt{2}}$

Let ${\tan }^{-1}(x)\in \left(-\frac{\pi }{2},\frac{\pi }{2}\right)$, for $x\in \mathbb{R}$. Then the number of real solutions of the equation $$\sqrt{1+\cos (2x)}=\sqrt{2}{\tan }^{-1}(\tan x)$$ in the set $\left(-\frac{3\pi }{2},-\frac{\pi }{2}\right)\cup \left(-\frac{\pi }{2},\frac{\pi }{2}\right)\cup \left(\frac{\pi }{2},\frac{3\pi }{2}\right)$ is equal to ____.

Let $A=\left\{\frac{1967+1686i\sin \theta }{7-3i\cos \theta }:\theta \in \mathbb{R}\right\}$. If $A$ contains exactly one positive integer $n$, then the value of $n$ is ____.

For any $y\in \mathbb{R}$, let ${\cot }^{-1}(y)\in (0,\pi )$ and ${\tan }^{-1}(y)\in \left(-\frac{\pi }{2},\frac{\pi }{2}\right)$. Then the sum of all the solutions of the equation ${\tan }^{-1}\left(\frac{6y}{9-y^2}\right)+{\cot }^{-1}\left(\frac{9-y^2}{6y}\right)=\frac{2\pi }{3}$ for $0<|y|<3$, is equal to

1. A.$2\sqrt{3}-3$
2. B.$3-2\sqrt{3}$
3. C.$4\sqrt{3}-6$
4. D.$6-4\sqrt{3}$