Trigonometry PYQ — Page 8

The value of ${\mathrm{cos}}^{2}10^{\circ}–\mathrm{cos}10^{\circ} \mathrm{cos}50^{\circ}+co{s}^{2}50^{\circ}$ is

If ${\mathrm{cos}}^{-1}(\frac{2}{3x})+{\mathrm{cos}}^{-1}(\frac{3}{4x})=\frac{\pi }{2} (x>\frac{3}{4}),$ then $x$ is equal to :

The value of $sin10^{\circ}sin30^{\circ}sin50^{\circ}sin70^{\circ}$ is:

The maximum value of $3\mathrm{cos}⁡\theta +5\mathrm{sin}⁡(\theta -\frac{\pi }{6})$ for any real value of $\theta$ is :

The value of $\mathrm{cos}⁡\frac{\pi }{{2}^{2}}\cdot \mathrm{cos}⁡\frac{\pi }{{2}^{3}}\cdot \ldots \cdot \mathrm{cos}⁡\frac{\pi }{{2}^{10}}\cdot \mathrm{sin}⁡\frac{\pi }{{2}^{10}}$ is:

If $x={sin}^{-1}(\mathrm{sin}⁡10)$ and $y={cos}^{-1} (\mathrm{cos}⁡10),$ then $y-x$ is equal to:

Let $f_{k}(x)=\frac{1}{k}\left(\sin ^{k} x+\cos ^{k} x\right)$ for $\mathrm{k}=1,2,3, \ldots$ Then for all $\mathrm{x} \in \mathrm{R},$ the value of $f_{4}(x)-f_{6}(x)$ is equal to :

Let $S={\theta \in [-2\pi ,2\pi ]:2{\mathrm{cos}}^{2}\theta +3\mathrm{sin}\theta =0}.$ Then the sum of the elements of $S$ is:

The value of ${\mathrm{sin}}^{-1}⁡(\frac{12}{13})-{\mathrm{sin}}^{-1}⁡(\frac{3}{5})$ is equal to:

The equation $y=sinx\mathrm{sin}⁡(x+2)-{\mathrm{sin}}^{2}⁡(x+1)$ represents a straight line lying in:

Considering only the principal values of inverse functions, the set $A={x\geq 0:{\mathrm{tan}}^{-1}(2x)+{\mathrm{tan}}^{-1}(3x)=\frac{\pi }{4}}$

The value of $\mathrm{cot}⁡(\sum _{n=1}^{19}{\mathrm{cot}}^{-1}⁡(1+\sum _{p=1}^{n}2p))$ is:

If ${\mathrm{cos}}^{-1}x-{\mathrm{cos}}^{–1}\frac{y}{2}=\alpha ,$ where $-1\leq x\leq 1,-2\leq y\leq 2,x\leq \frac{y}{2},$ then for all $x,y,4{x}^{2}-4xy\mathrm{cos}\alpha +{y}^{2}$ is equal to :

If $\tan A$ and $\tan B$ are the roots of the quadratic equation, $3 x^2-10 x-25=0$ then the value of $3 \sin ^2(A+B)-10 \sin (A+B) \cdot \cos (A+B)-25 \cos ^2$ $(A+B)$ is

The number of solutions of $\sin 3 x=\cos 2 x$, in the interval $\left(\frac{\pi}{2}, \pi\right)$ is

If sum of all the solutions of the equation $8\mathrm{cos}x\cdot (\mathrm{cos}(\frac{\pi }{6}+x)\cdot \mathrm{cos}(\frac{\pi }{6}-x)-\frac{1}{2})=1$ in $[0, \pi ]$ is $k\pi$, then $k$ is equal to:

The lengths of two adjacent sides of a cyclic quadrilateral are $2$ units and $5$ units and the angle between them is ${60}^{o}$. If the area of the quadrilateral is $4\sqrt{3}$ sq. units, then the perimeter of the quadrilateral is

A value of $x$ satisfying the equation $\mathrm{sin}⁡[{\mathrm{cot}}^{-1}⁡(1+x)]=\mathrm{cos}⁡[{\mathrm{tan}}^{-1}⁡x],$ is:

The value of ${\mathrm{tan}}^{-1}⁡[\frac{\sqrt{1+{x}^{2}}+ \sqrt{1-{x}^{2}}}{\sqrt{1+{x}^{2}}- \sqrt{1-{x}^{2}}}],$ $|x|<\frac{1}{2}, x\neq 0,$ is equal to:

If $5({\mathrm{tan}}^{2}⁡x-{\mathrm{cos}}^{2}⁡x)=2\mathrm{cos}⁡ 2x+9,$ then the value of $\mathrm{cos}⁡4x$ is

The number of $x \in [0, 2\pi ]$ for which $|\sqrt{2{\mathrm{sin}}^{4}x+18{\mathrm{cos}}^{2}x}- \sqrt{2{\mathrm{cos}}^{4}x+18{\mathrm{sin}}^{2}x}|=1$ is:

If $m$ and $M$ are the minimum and the maximum values of $4+\frac{1}{2}{\mathrm{sin}}^{2}⁡2x-2{\mathrm{cos}}^{4}⁡x, x \in R,$ then $M-m$ is equal to:

Let $P={\theta :\mathrm{sin}\theta -\mathrm{cos}\theta =\sqrt{2}\mathrm{cos}\theta }$ and $Q={\theta :\mathrm{sin}\theta +\mathrm{cos}\theta =\sqrt{2}\mathrm{sin}\theta },$ be two sets. Then

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