Maximum and Minimum Values of Trigonometric Functions

If $\sin x+\cos x=\sqrt{y+\frac{1}{y}}, x\in [0,\pi ]$, then

For any real $\theta$, the maximum value of ${\cos }^2\left(\cos \theta \right)+{\sin }^2\left(\sin \theta \right)$ is 
 

The minimum value of $3{\tan }^2\theta +12{\cot }^2\theta$ is

The value of $a$ for which the equation; $4{cosec}^2\left(\pi \right(a+x\left)\right)+a^2-4a=0$ has a real solution is:

If  $0<\alpha <\beta <\gamma <\frac{\mathrm{\pi }}{2}$ then, $\frac{\sin \alpha +\sin \beta +\sin \gamma }{\cos \alpha +\cos \beta +\cos \gamma }$ lies between:
 

The maximum value of the expression, $\left|\sqrt{{\sin }^{2}x+2a^2}-\sqrt{2a^2-1-{\cos }^{2}x}\right|$, where $a$ and $x$ are real numbers is

Determine all the values of $x$ in the interval $x\in [0, 2\text{π}]$ which satisfy the inequality $2\cos x\le \left|\sqrt{1+\sin 2x}-\sqrt{1-\sin 2x}\right|\le \sqrt{2}$.

Prove that $\frac{1}{3}\le \frac{{\sec }^4\theta -3{\tan }^2\theta }{{\sec }^4\theta -{\tan }^2\theta }<1$.

Find the set of values of $\alpha$ for which the point $\left(\sin \alpha , \cos \alpha \right)$ does not lie outside the curve $2y^2+x-2=0$ in the interval $\left[\frac{\pi }{2}, \frac{3\pi }{2}\right]$.

Four real constants $a$, $b$, $A$, $B$ are given and $f\left(\theta \right)=1-a\cos \theta -b\sin \theta -A\cos 2\theta -B\sin 2\theta .$  Prove that if $f\left(\theta \right)\ge 0,\forall \theta \in R$ then $a^2+b^2\le 2$ and $A^2+B^2\le 1$.

If the quadratic equation $4^{{\sec }^2\alpha }{x}^2+2x+\left({\beta }^2-\beta +\frac{1}{2}\right)=0$ have real roots, then find all the possible values of $\cos \alpha +{\cos }^{-1}\beta$.

If $a$, $b$, $c$ and $k$ are constant quantities and $\alpha , \beta , \gamma$ are variable subject to the relation $a\tan \alpha +b\tan \beta +c\tan \gamma =k,$ find the minimum value of ${\tan }^2\alpha +{\tan }^2\beta +{\tan }^2\gamma$.

If $a$ and $b$ are positive quantities and $a\ge b,$ find the minimum positive values of $a\sec \theta -b\tan \theta$.

Find all the solutions of the equation $\sin \left(x-\frac{\pi }{4}\right)-\cos \left(x+\frac{3\pi }{4}\right)=1$which satisfy, $\frac{2\cos 7x}{\cos 3+\sin 3}>2^{\cos 2x},\forall x\in \left[0,\pi \right]$.

Find the maximum and minimum values for $3\cos x+5\sin \left(x-\frac{\mathrm{\pi }}{6}\right)$.

Find the maximum and minimum values for $2\sin x+4\cos x+3$.

Find the maximum value of $4{\sin }^{2}x+3{\cos }^{2}x+\sin \left(\frac{x}{2}\right)+\cos \left(\frac{x}{2}\right)$.
 

Find the maximum and minimum values for $\cos \left(\cos x\right)$.