Trigonometry

If $\theta \in \left[-\frac{7\pi}{6}, \frac{4\pi}{3}\right]$, then the number of solutions of \\ $\sqrt{3}\csc^2\theta - 2(\sqrt{3} - 1)\csc\theta - 4 = 0$, is equal to:

If $\theta \in [-2\pi, 2\pi]$, then the number of solutions of $2\sqrt{2}\cos^2\theta + (2-\sqrt{6})\cos\theta - \sqrt{3} = 0$ is equal to:

The number of solutions of the equation $(4-\sqrt{3})\sin x - 2\sqrt{3}\cos^2x = -\frac{4}{1+\sqrt{3}},\quad x\in[-2\pi,\tfrac{5\pi}{2}]$ is equal to:

The number of solutions of the equation $2x + 3\tan x = \pi$, $x\in[-2\pi,2\pi]\setminus\{\pm\tfrac{\pi}{2},\pm\tfrac{3\pi}{2}\}$ is:

The sum of the infinite series $\cot^{-1}\left(\tfrac{7}{4}\right)+\cot^{-1}\left(\tfrac{19}{4}\right)+\cot^{-1}\left(\tfrac{39}{4}\right)+\cot^{-1}\left(\tfrac{67}{4}\right)+\ldots$ is:

Considering the principal values of the inverse trigonometric functions, \sin⁻¹\bigl(\tfrac{\sqrt{3}}{2}x + \tfrac{1}{2}\sqrt{1-x²}\bigr), -\tfrac{1}{2} < x < \tfrac{1}{\sqrt{2}}, is equal to:

If 10\sin⁴\theta + 15\cos⁴\theta = 6, then the value of \displaystyle \frac{27\csc⁶\theta + 8\sec⁶\theta}{16\sec⁸\theta} is:

The number of solutions of the equation $\cos 2\theta\cos\frac{\theta}{2} + \cos\frac{5\theta}{2} = 2\cos^3\frac{5\theta}{2}$ in $[-\tfrac{\pi}{2},\tfrac{\pi}{2}]$ is:

If for $\theta \in \left[ -\frac{\pi}{3}, 0 \right]$, the points $(x, y) = \left( 3 \tan\left(\theta + \frac{\pi}{3}\right), 2 \tan\left(\theta + \frac{\pi}{6}\right) \right)$ lie on $xy + \alpha x + \beta y + \gamma = 0$, then $\alpha^2 + \beta^2 + \gamma^2$ is equal to:

The value of $\cot^{-1} \left( \frac{\sqrt{1 + \tan^2(2)} - 1}{\tan(2)} \right) - \cot^{-1} \left( \frac{\sqrt{1 + \tan^2\left(\frac{1}{2}\right)} + 1}{\tan\left(\frac{1}{2}\right)} \right)$ is equal to: