Trigonometric Equations

The value of $4{\cot }^{2}45°–{\sec }^{2}60°+{\sin }^{2}30°$ is ____.

The value of $\cos 10º-\sin 10°$ is______?

If  $-\frac{\pi }{2}\le x\le \frac{\pi }{2}$, then number of the points of intersection of the given curves $y=\cos x$, $y=\sin 3x$ is/are

If  $\tan \left(\pi \cos \theta \right)=\cot (\pi \sin \theta ),$ then $\sin \left(\theta +\frac{\pi }{4}\right)$ can be equal to

The smallest positive root of the equation, $\tan x-x=0$ lies in

If  $x\in \left(-\pi \text{, }\pi \right)$ then find the number of solutions of the equation  $2^{1+\left|\cos x\right|+\left|{\cos }^{2}x\right|+\mathrm{...}}\mathrm{}$ = 4

If  $\text{x}\in \left(-\pi \text{, }\pi \right)$ then the number of solutions of the equation  $2^{1+\left|\cos x\right|+\left|{\cos }^{2}x\right|+\mathrm{...}}\mathrm{}$ = 4 is/are-

The number of all possible value of  $\theta$ , where  $0<\theta <\pi$ , for which the system of equations 

$(y+z)\cos 3\theta =\left(xyz\right)\sin 3\theta$

$x\sin 3\theta =\frac{2\cos 3\theta }{y}+\frac{2\sin 3\theta }{z}$

and  $\left(xyz\right)\sin 3\theta =(y+2z)\cos 3\theta +y\sin 3\theta$ have a solution  $(x_0,y_0,z_0)$ with  $y_0{z}_0\ne 0$ , is/are

The number of solutions of the equation $\sin \theta +\cos \theta =\sin 2\theta$ in the interval $\left[-\pi ,\pi \right]$ is

Find the domain of trigonometric equation $y=\sqrt{\tan x}$ in $\left[0,2\mathrm{\pi }\right]$.

Find the domain of trigonometric equation $y=\sqrt{\mathrm{cosec}x}$ in $\left[0,2\mathrm{\pi }\right]$.

Find the domain of trigonometric equation $y=\sqrt{\sec x}$ in $\left[0,2\mathrm{\pi }\right]$.

Find the domain of trigonometric equation $y=\sqrt{\cos x}$ in $\left[0,2\mathrm{\pi }\right]$.

General solutions of $\sin x=-\frac{\sqrt{3}}{2}$ and $\cos x=\frac{-1}{2}$ are equal.