Problems Based on Extreme Values of Trigonometric Functions

Show that the equation $\sec \theta +cosec\theta =c$ has two roots between $0$ and $2\pi ,$ if $c^2<8$ and four roots$\text{ if }{c}^2>8$.

Find all number pairs $x, y$ that satisfy the equation: ${\tan }^{4}x+{\tan }^{4}y+2{\cot }^{2}x·{\cot }^{2}y=3+{\sin }^2(x+y)$

Find all the values of $a(a\ne 0)$ for which the equation: ${\int }_0^x\left(t^2-8t+13\right)dt=x\sin \frac{a}{x}$ has a solution. Find that solution.

Find all the solutions of$x, y$ in the equation  $4\left(3\sqrt{4x-x^2}{\sin }^2\left(\frac{x+y}{2}\right)+2\cos \left(x+y\right)\right)=13+4{\cos }^2\left(x+y\right)$

Solve for $x$ and $y:$

$12\sin x+5\cos x=2y^2-8y+21$

Find $k$ if the equation $2\cos x+\cos 2kx=3$ has only one solution.

Find the set of all $x$ in $(-\pi ,\pi )$ satisfying $|4\sin x-1|<\sqrt{5}$.