If $2 \tan^{-1}(\cos x) = \tan^{-1}(2 \operatorname{cosec} x)$,then the value of $x$ is

If $\tan ^{-1}\left[\frac{\sqrt{1+x^2}-\sqrt{1-x^2}}{\sqrt{1+x^2}+\sqrt{1-x^2}}\right]=\alpha$,then the value of $\sin 2 \alpha$ is

If $\cos \left(\cos ^{-1} \frac{\sqrt{3}}{2}+\sin ^{-1} x\right)=1$,then find the value of $x$.

The value of $\sin \left(\cos ^{-1}\left(-\frac{1}{3}\right)-\sin ^{-1}\left(\frac{1}{3}\right)\right)$ is

Let $a_1=1, a_2, a_3, a_4, \ldots$ be consecutive natural numbers. Then $\tan ^{-1}\left(\frac{1}{1+ a _1 a _2}\right)+\tan ^{-1}\left(\frac{1}{1+ a _2 a _3}\right)+\ldots+\tan ^{-1}\left(\frac{1}{1+ a _{2021} a _{2022}}\right)$ is equal to

If $\sin ^{-1}(4 x)+\sin ^{-1}(4 \sqrt{3} x)=-\frac{\pi}{2}$,then the absolute value of $x$ is