Trigonometry PYQ — Page 9

If $0 \leq x<2\pi ,$ then the number of real values of $x,$ which satisfy the equation $\mathrm{cos}x+\mathrm{cos}2x+\mathrm{cos}3x+\mathrm{cos}4x=0,$ is

If $f(x)=2{\mathrm{tan}}^{-1}x+{\mathrm{sin}}^{-1}(\frac{2x}{1+{x}^{2}}), x>1$, then $f(5)$ is equal to

If $\mathrm{cos}\alpha +\mathrm{cos}\beta =\frac{3}{2} \text{and} \mathrm{sin}\alpha +\mathrm{sin}\beta =\frac{1}{2}$and $\theta$ is the arithmetic mean of $\alpha &\beta ,$ then $\mathrm{sin}2\theta +\mathrm{cos}2\theta$ is equal to:

Let ${\mathrm{tan}}^{-1}y={\mathrm{tan}}^{-1}x+{\mathrm{tan}}^{-1}(\frac{2x}{1-{x}^{2}})$, where $|x|<\frac{1}{\sqrt{3}}$,Then a value of $y$ is

In a $\Delta ABC$, $\frac{a}{b}=2+\sqrt{3}$, and $\angle C=60^{\circ}.$ Then the ordered pair $(\angle A,\angle B)$ is equal to:

If $2 \cos \theta+\sin \theta=1\left(\theta \neq \frac{\pi}{2}\right)$, then $7 \cos \theta+6 \sin \theta$ is equal to:

The principal value of ${\mathrm{tan}}^{-1}(cot\frac{43\pi }{4})$ is

Statement I: The equation $\left(\sin ^{-1} \mathrm{x}\right)^3+$ $\left(\cos ^{-1} \mathrm{x}\right)^3-\mathrm{a} \pi^3=0$ has a solution for all $\mathrm{a} \geq \frac{1}{32}$. Statement II: For any $\mathrm{x} \in \mathrm{R}$, $\sin ^{-1} x+\cos ^{-1} x=\frac{\pi}{2}$ and $0 \leq\left(\sin ^{-1} x-\frac{\pi}{4}\right)^2 \leq \frac{9 \pi^2}{16}$

Let ${f}_{k}(x)=\frac{1}{k}({\mathrm{sin}}^{k}x+{\mathrm{cos}}^{k}x)$ where $x\in R$ and $k\geq 1$. Then ${f}_{4}(x)-{f}_{6}(x)$ equals

If $\text{cosec }\theta =\frac{\text{p}+\text{q}}{\text{ p}-\text{q }}(\text{p}\neq \text{q, p}\neq 0),$then $|\text{cot}(\frac{\pi }{4}+\frac{\theta }{2})|$ is equals to:

The number of solutions of the equation $\sin 2 x-2 \cos x+4 \sin x=4$ in the interval $[0,5 \pi]$ is :

The number of solutions of the equation, $\sin ^{-1} x=2 \tan ^{-1} x$ (in principal values) is :

A value of $x$ for which $\sin \left(\cot ^{-1}(1+x)\right)=\cos$ $\left(\tan ^{-1} x\right)$, is :

The expression $\frac{\mathrm{tanA}}{1-\mathrm{cotA}}+\frac{\mathrm{cotA}}{1-\mathrm{tanA}}$ can be written as :

$S=\tan ^{-1}\left(\frac{1}{n^2+n+1}\right)+\tan ^{-1}\left(\frac{1}{n^2+3 n+3}\right)+\ldots$ $+\tan ^{-1}\left(\frac{1}{1+(n+19)(n+20)}\right)$, then $\tan S$ is equal to :

Let $S=\left\{\left(\begin{array}{ll}a_{11} & a_{12} \\ a_{21} & a_{22}\end{array}\right): a_{i j} \in\{0,1,2\}, a_{11}=a_{22}\right\}$ Then the number of non-singular matrices in the set $S$ is :

Let $\mathrm{A}=\{\theta: \sin (\theta)=\tan (\theta)\}$ and $\mathrm{B}=(\theta: \cos (\theta)=$ 1\} be two sets. Then:

The value of $\cos 255^{\circ}+\sin 195^{\circ}$ is

Suppose $\theta$ and $\phi(\neq 0)$ are such that $\sec (\theta+\phi)$, sec $\theta$ and $\sec (\theta-\phi)$ are in A.P. If $\cos \theta=k \cos \left(\frac{\phi}{2}\right)$ for some $k$, then $k$ is equal to

A value of $\tan ^{-1}\left(\sin \left(\cos ^{-1}\left(\sqrt{\frac{2}{3}}\right)\right)\right)$ is

If $A=\sin ^2 x+\cos ^4 x$, then for all real $x$

Let $\cos (\alpha+\beta)=\frac{4}{5}$ and let $\sin (\alpha-\beta)=\frac{5}{13}$, where $0 \leq \alpha, \beta \leq \frac{\pi}{4}$, then $\tan 2 \alpha=$

The value of $\cot \left(\operatorname{cosec}^{-1} \frac{5}{3}+\tan ^{-1} \frac{2}{3}\right)$ is

If $\sin ^{-1}\left(\frac{x}{5}\right)+\operatorname{cosec}^{-1}\left(\frac{5}{4}\right)=\frac{\pi}{2}$ then a value of $x$ is