Find the value of $\tan ^{-1}\left(\tan \frac{5 \pi}{6}\right)+\cos ^{-1}\left(\cos \frac{13 \pi}{6}\right)$.
Evaluate $\cos \left[\cos ^{-1}\left(\frac{-\sqrt{3}}{2}\right)+\frac{\pi}{6}\right]$.
Prove that $\cot \left(\frac{\pi}{4}-2 \cot ^{-1} 3\right)=7$.
Find the value of $\tan ^{-1}\left(-\frac{1}{\sqrt{3}}\right)+\cot ^{-1}\left(\frac{1}{\sqrt{3}}\right)+\tan ^{-1}\left[\sin \left(\frac{-\pi}{2}\right)\right]$.
Find the value of $\tan ^{-1}\left(\tan \frac{2 \pi}{3}\right)$.
Show that $2 \tan ^{-1}(-3)=\frac{-\pi}{2}+\tan ^{-1}\left(\frac{-4}{3}\right)$.
Find the real solution of $$\tan ^{-1} \sqrt{x(x+1)}+\sin ^{-1} \sqrt{x^2+x+1}=\frac{\pi}{2} .$$
Find the value of $\sin \left(2 \tan ^{-1} \frac{1}{3}\right)+\cos \left(\tan ^{-1} 2 \sqrt{2}\right)$.
If $2 \tan ^{-1}(\cos \theta)=\tan ^{-1}(2 \operatorname{cosec} \theta)$, then show that $\theta=\frac{\pi}{4}$, where $n$ is any integer.
Show that $\cos \left(2 \tan ^{-1} \frac{1}{7}\right)=\sin \left(4 \tan ^{-1} \frac{1}{3}\right)$.
Solve the equation $\cos \left(\tan ^{-1} x\right)=\sin \left(\cot ^{-1} \frac{3}{4}\right)$.
Prove that $\tan ^{-1}\left(\frac{\sqrt{1+x^2}+\sqrt{1-x^2}}{\sqrt{1+x^2}-\sqrt{1-x^2}}\right)=\frac{\pi}{4}+\frac{1}{2} \cos ^{-1} x^2.$
Find the simplified form of $$ \cos ^{-1}\left(\frac{3}{5} \cos x+\frac{4}{5} \sin x\right) \text {, where } x \in\left[\frac{-3 \pi}{4}, \frac{\pi}{4}\right] $$
Prove that $\sin ^{-1} \frac{8}{17}+\sin ^{-1} \frac{3}{5}=\sin ^{-1} \frac{77}{85}$.
Show that $\sin ^{-1} \frac{5}{13}+\cos ^{-1} \frac{3}{5}=\tan ^{-1} \frac{63}{16}$.
Prove that $\tan ^{-1} \frac{1}{4}+\tan ^{-1} \frac{2}{9}=\sin ^{-1} \frac{1}{\sqrt{5}}$.
Find the value of $4 \tan ^{-1} \frac{1}{5}-\tan ^{-1} \frac{1}{239}$.
Show that $\tan \left(\frac{1}{2} \sin ^{-1} \frac{3}{4}\right)=\frac{4-\sqrt{7}}{3}$ and justify why the other value $\frac{4+\sqrt{7}}{3}$ is ignored?
If $a_1, a_2, a_3, \ldots, a_n$ is an arithmetic progression with common difference $d$, then evaluate the following expression.
$$\begin{aligned} \tan \left[\tan ^{-1}\left(\frac{d}{1+a_1 a_2}\right)+\tan ^{-1}\left(\frac{d}{1+a_2 a_3}\right)\right. & +\tan ^{-1}\left(\frac{d}{1+a_3 a_4}\right) \\ & \left.+\ldots+\tan ^{-1}\left(\frac{d}{1+a_{n-1} a_n}\right)\right] \end{aligned}$$
Which of the following is the principal value branch of $\cos ^{-1} x$ ?
Which of the following is the principal value branch of $\operatorname{cosec}^{-1} x$ ?
If $3 \tan ^{-1} x+\cot ^{-1} x=\pi$, then $x$ equals to
The value of $\sin ^{-1}\left[\cos \left(\frac{33 \pi}{5}\right)\right]$ is
The domain of the function $\cos ^{-1}(2 x-1)$ is
The domain of the function defined by $f(x)=\sin ^{-1} \sqrt{x-1}$ is
If $\cos \left(\sin ^{-1} \frac{2}{5}+\cos ^{-1} x\right)=0$, then $x$ is equal to
The value of $\sin \left[2 \tan ^{-1}(0.75)\right]$ is
The value of $\cos ^{-1}\left(\cos \frac{3 \pi}{2}\right)$ is
The value of $2 \sec ^{-1} 2+\sin ^{-1}\left(\frac{1}{2}\right)$ is
If $\tan ^{-1} x+\tan ^{-1} y=\frac{4 \pi}{5}$, then $\cot ^{-1} x+\cot ^{-1} y$ equals to
If $\sin ^{-1}\left(\frac{2 a}{1+a^2}\right)+\cos ^{-1}\left(\frac{1-a^2}{1+a^2}\right)=\tan ^{-1}\left(\frac{2 x}{1-x^2}\right)$, where $\left.a, x \in\right] 0,1[$, then the value of $x$ is
The value of $\cot \left[\cos ^{-1}\left(\frac{7}{25}\right)\right]$ is
The value of $\tan \left(\frac{1}{2} \cos ^{-1} \frac{2}{\sqrt{5}}\right)$ is
If $|x| \leq 1$, then $2 \tan ^{-1} x+\sin ^{-1}\left(\frac{2 x}{1+x^2}\right)$ is equal to
If $\cos ^{-1} \alpha+\cos ^{-1} \beta+\cos ^{-1} \gamma=3 \pi$, then $\alpha(\beta+\gamma)+\beta(\gamma+\alpha)+\gamma(\alpha+\beta)$ equals to
The number of real solutions of the equation $\sqrt{1+\cos 2 x}=\sqrt{2} \cos ^{-1}(\cos x)$ in $\left[\frac{\pi}{2}, \pi\right]$ is
If $\cos ^{-1} x>\sin ^{-1} x$, then
The principal value of $\cos ^{-1}\left(-\frac{1}{2}\right)$ is ........... .
The value of $\sin ^{-1}\left(\sin \frac{3 \pi}{5}\right)$ is ........ .
If $\cos \left(\tan ^{-1} x+\cot ^{-1} \sqrt{3}\right)=0$, then the value of $x$ is ............ .
The set of values of $\sec ^{-1} \frac{1}{2}$ is ........... .
The principal value of $\tan ^{-1} \sqrt{3}$ is ............ .
The value of $\cos ^{-1}\left(\cos \frac{14 \pi}{3}\right)$ is ............. .
The value of $\cos \left(\sin ^{-1} x+\cos ^{-1} x\right)$, where $|x| \leq 1$, is ............ .
The value of $\tan \left(\frac{\sin ^{-1} x+\cos ^{-1} x}{2}\right)$, when $x=\frac{\sqrt{3}}{2}$, is ............... .
If $y=2 \tan ^{-1} x+\sin ^{-1}\left(\frac{2 x}{1+x^2}\right)$, then $\ldots \ldots \ldots< y<$ .............. .
The result $\tan ^{-1} x-\tan ^{-1} y=\tan ^{-1}\left(\frac{x-y}{1+x y}\right)$ is true when the value of $x y$ is ............ .
The value of $\cot ^{-1}(-x) x \in R$ in terms of $\cot ^{-1} x$ is ............ .
All trigonometric functions have inverse over their respective domains.
The value of the expression $\left(\cos ^{-1} x\right)^2$ is equal to $\sec ^2 x$.
The domain of trigonometric functions can be restricted to any one of their branch (not necessarily principal value) in order to obtain their inverse functions.
The least numerical value, either positive or negative of angle $\theta$ is called principal value of the inverse trigonometric function.
The graph of inverse trigonometric function can be obtained from the graph of their corresponding function by interchanging $X$ and $Y$-axes.
The minimum value of $n$ for which $\tan ^{-1} \frac{n}{\pi}>\frac{\pi}{4}, n \in N$, is valid is 5 .
The principal value of $\sin ^{-1}\left[\cos \left(\sin ^{-1} \frac{1}{2}\right)\right]$ is $\frac{\pi}{3}$.