Trigonometric Equations

If $\left(\mathrm{cosec} \mathrm{\theta }-\sin \mathrm{\theta }\right)={\mathrm{a}}^3 \& \left(\sec \mathrm{\theta }-\cos \mathrm{\theta }\right)={\mathrm{b}}^3$ then, the value of ${\mathrm{a}}^2{\mathrm{b}}^2\left({\mathrm{a}}^2+{\mathrm{b}}^2\right)$ is _______.

Maximum value of $(2\sin \theta +3\cos \theta )$ is _____.

What is the maximum value of $\sin 3i\times \cos 2i+\cos 3i\times \sin 2i$?

What is the value of $\left(1+\cot \mathrm{x}-\mathrm{cosec} \mathrm{x}\right)\left(1+\tan \mathrm{x}+\sec \right)$?

If $\mathrm{tanA}=\frac{1-\mathrm{cosB}}{\mathrm{sinB}}$, then what is$\frac{2\mathrm{tanA}}{1-{\tan }^2\mathrm{A}}$  equal to ______.

If $\sin (3\mathrm{\alpha }-\mathrm{\beta })=1$ and $\cos (2\mathrm{\alpha }+\mathrm{\beta })=\frac{1}{2}$, then the value of$\tan \mathrm{\alpha }$is ______.

If $\frac{\sec \theta +\tan \theta }{\sec \theta -\tan \theta }=\frac{5}{3}$ then $\sin A$, equals to-where $\mathrm{\theta }=\mathrm{A}$?

If $\theta$ is a positive acute angle and $\tan 2\theta ·\tan 3\theta =1$, then the value of $\left(2 {\cos }^2\frac{5\theta }{2}-1\right)$ is _________.

If $8\sin x=4+\cos x$, the values of $\sin x$ are:

If $\cos x+{\cos }^{2}x=1$, the numerical value of $\left({\sin }^{12}x+3 {\sin }^{10}x+3 {\sin }^{8}x+{\sin }^{6}x-1\right)$ is ________.

If $\sec \mathrm{\alpha }+\tan \mathrm{\alpha }=2$, then the value of $\sin \alpha$ is _________.

If $\tan (x+y) \tan (x-y)=1$ then the value of $\tan x$.

The value of $\sin i$ and $\cos i$ is always less than $1$ in $0<i<90°$. Then the value of $\sin i+\cos i$ is _____.

If $\sec \mathrm{x}+\cos \mathrm{x}=3$, what will be the value of ${\tan }^2\mathrm{x}-{\sin }^2\mathrm{x}$ ?

If $\sin \mathrm{\theta }+\cos \mathrm{\theta }=\mathrm{p}$ and $\sec \mathrm{\theta }+\mathrm{cosec} \mathrm{\theta }=\mathrm{q}$, then the value of $\mathrm{q}({\mathrm{p}}^2-1)$ is _____.

If $\sin 17°=\frac{\mathrm{x}}{\mathrm{y}}$, then the value of $\sec 17°-\sin 73°$ is _____.

If${\tan }^{2}y {\mathrm{cosec}}^{2}x–1={\tan }^{2}y$, then which one of the following is correct?

If $\tan \theta +\tan \varphi =x, cot \theta +cot \varphi =y$, then what is the value of $cot \left(\theta +\varphi \right)?$

If $x = \mathrm{a} \sec \mathrm{q}+ \mathrm{b} \tan \mathrm{q}$, $y= \mathrm{b} \sec \mathrm{q} + \mathrm{a} \tan \mathrm{q}$, then $x^2-y^2$ is ________.

If $3\sin \mathrm{x}+5\cos \mathrm{x}=5$, then what is the value of $(3\cos \mathrm{x}-5\sin \mathrm{x})$ ______.