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 _______.

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

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 $\cos x+{\cos }^{2}x=1$, 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 $\tan x$.

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

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 $r \sin \theta =\frac{7}{2} \& r \cos \theta =\frac{7\sqrt{3}}{2}$, then the values of $r \& \theta$ are ________.

The value of $\sin {10}^{°} \sin {30}^{°} \sin {50}^{°} \sin {70}^{°}$ is  _______.

If $A$ and $B$ are acute angle and $\sin (A-B)=\frac{1}{2}$ and $\cos (A+B)=\frac{1}{2}$ then $A$ and $B$ will be_____

Find the value of:

$4\left({\sin }^6\theta +{\cos }^6\theta \right)-6\left({\sin }^4\theta +{\cos }^4\theta \right)$

The eliminant of $\theta$ from $x \cos \theta -y \sin \theta =2$ and $x \sin \theta +y \cos \theta =4$ will give __________.

If $x=3 \cos A \cos B, y=3 \cos A \sin B \& z=3 \sin A$ then, $x^2+y^2+z^2$ is _______.

If $\cos \left(\alpha +\beta \right)=\frac{4}{5}$ and $\sin \left(\alpha -\beta \right)=\frac{5}{13}$, where $\alpha , \beta$ lies between $0 \& \frac{\pi }{4}$, then the value of $\tan 2\alpha$ is __________.

If $m=\mathrm{cosec}x-\sin x$ and $n=\sec x-\cos x$ then $\tan x$ is equal to _____.

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

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