JEE Main 2019 — Trigonometry

All $x$ satisfying the inequality $\left(\cot ^{-1} x\right)^{2}-7\left(\cot ^{-1} x\right)+10>$ 0 , lie in the interval :

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

For any $\theta \in (\frac{\pi }{4},\frac{\pi }{2}),$ the expression $3{(\mathrm{sin}\theta -\mathrm{cos}\theta )}^{4}+6{(\mathrm{sin}\theta +\mathrm{cos}\theta )}^{2}+4 {sin}^{6}\theta$ equals:

If $cos(\alpha +\beta )=\frac{3}{5} ,\mathrm{sin}⁡(\alpha -\beta )=\frac{5}{13}$ and $0<\alpha ,\beta <\frac{\pi }{4},$ then $\mathrm{tan}⁡(2\alpha )$ is equal to:

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

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 :

If $0\leq x<\frac{\pi }{2},$ then the number of values of $x$ for which $\mathrm{sin}x-\mathrm{sin}2x+\mathrm{sin}3x=0,$ 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 $\alpha ={cos}^{-1}(\frac{3}{5})$ , $\beta ={tan}^{-1}(\frac{1}{3})$ , where $0<\alpha ,\beta <\frac{\pi }{2},$ then $\alpha -\beta$ is equal to

Let $S$ be the set of all $\alpha \in R$ such that the equation, $cos2x+\alpha sinx=2\alpha -7$ has a solution. Then $S$ 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 equation $y=sinx\mathrm{sin}⁡(x+2)-{\mathrm{sin}}^{2}⁡(x+1)$ represents a straight line lying in:

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

The number of solutions of the equation $1+{\mathrm{sin}}^{4}x={\mathrm{cos}}^{2}3x,x\in [-\frac{5\pi }{2},\frac{5\pi }{2}]$ is:

The sum of all values of $\theta \in (0, \frac{\pi }{2})$ satisfying ${\mathrm{sin}}^{2}2\theta +{\mathrm{cos}}^{4}2\theta =\frac{3}{4}$ is

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

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

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

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