Solving Simultaneous Equations

If $A$ and $B$ are acute angles satisfying the equations $3{\sin }^{2}A+2{\sin }^{2}B=1$ and $3\sin 2A-2\sin 2B=0,$ then $(A+2B)$ is 

The number of pairs $(x, y)$ satisfy  equations $\sin x+\sin y=\sin (x+y)$ and $\left|x\right|+\left|y\right|=1,$ is

The most general solution of $\tan \theta =-1$ and $\cos \theta =\frac{1}{\sqrt{2}}$ is

The number of points inside or on the circle $x^2+y^2=4$ satisfying ${\tan }^{4}x+{\cot }^{4}x+1=3{\sin }^{2}y,$ is:

The number of solutions of the equation $\left|\sin \theta \right|=\left|\cos 3\theta \right|$ in $[-2\pi ,2\pi ]$ is

The number of real solutions for $x, y$ if $y=\left|\sin x\right|$ and $y={\sin }^{-1}\left(\sin x\right)$ where $x\in \left[-2\pi ,2\pi \right]$, is

Total number of roots of the equation $3^{\cos x}=\left|\sin x\right|$, belonging to $\left[-2\pi ,2\pi \right]$ are

If $\sin 2\theta =\cos 3\theta$, then the number of elements for the set $\theta$ in $0\le \theta \le 2\pi$

Determine the range of values of $\theta \in \left[0,2\pi \right]$ for which $\left(\cos \theta ,\sin \theta \right)$ lies inside the triangle formed by $x+y-2=0, x-y-1=0$ and $6x+2y-\sqrt{10}=0$

Solve the system of equations  ${\tan }^{2}x+{\cot }^{2}x=2{\cos }^{2}y$ and  ${\cos }^{2}y+{\sin }^{2}z=1$

Solve for $x$ and $y:$   $4^{\sin x}+3^{\frac{1}{\cos y}}=11$
    $5·{16}^{\sin x}-2·3^{\frac{1}{\cos y}}=2$

Solve for $x$: $\left|\cos x\right|=\cos x-2\sin x$

Find the number of solutions of the equation: $\left|\cos x\right|=\left[x\right]$ (where $[.]$ denotes the greatest integer function).

Find the number of solutions of the equations: $2^{\cos x}=\left|\sin x\right|,\text{ when }x\in [-2\pi ,2\pi ]$

Solve the equation for $x$ and $y$; $|\mathrm{s}\mathrm{i}\mathrm{n}x+\mathrm{c}\mathrm{o}\mathrm{s}x{|}^{{\mathrm{s}\mathrm{i}\mathrm{n}}^{2}x-\frac{1}{4}}=1+\left|\mathrm{s}\mathrm{i}\mathrm{n}y\right|$ and ${\mathrm{c}\mathrm{o}\mathrm{s}}^{2}y=1+{\mathrm{s}\mathrm{i}\mathrm{n}}^{2}y$.