Trigonometry PYQ — Page 10

The number of values of $x$ in the interval $[0,3 \pi]$ satisfying the equation $2 \sin ^2 x+5 \sin x-3=0$ is

If $0 < x < \pi$ and $\cos x+\sin x=\frac{1}{2}$, then $\tan x$ is

If $\cos ^{-1} x-\cos ^{-1} \frac{y}{2}=\alpha$, then $4 x^2-4 x y \cos \alpha+y^2$ is equal to

Let $\alpha, \beta$ be such that $\pi < \alpha-\beta < 3 \pi$. If $\sin \alpha+\sin \beta=-\frac{21}{65}$ and $\cos \alpha+\cos \beta=-\frac{27}{65}$, then the value of $\cos \frac{\alpha-\beta}{2}$ is

The trigonometric equation $\sin ^{-1} x=2 \sin ^{-1}$ a has a solution for

The period of $\sin ^2 \theta$ is

$\cot ^{-1}(\sqrt{\cos \alpha})=\tan ^{-1}(\sqrt{\cos \alpha})=x$, then $\sin x=$

The number of solution of $\tan x+\sec x=2 \cos x$ in $[0,2 \pi)$ is