Inverse Trigonometric Functions · Mathematics · MHT CET

If $x=\tan ^{-1}\left\{\frac{\sqrt{1+t^2}-1}{t}\right\}, y=\cos ^{-1}\left\{\frac{1-t^2}{1+t^2}\right\}, \quad$ then $\frac{\mathrm{d} y}{\mathrm{~d} x}$ is equal to

$$ \cot ^{-1}\left(2 \cdot 1^2\right)+\cot ^{-1}\left(2 \cdot 2^2\right)+\cot ^{-1}\left(2 \cdot 3^2\right)+\ldots \ldots \ldots \infty= $$

The value of $\tan \left[2 \tan ^{-1} \frac{1}{5}-\frac{\pi}{4}\right]$ is

If $\cot ^{-1}(\sqrt{\cos \alpha})-\tan ^{-1}(\sqrt{\cos \alpha})=x$, then the value of $\sin x$ is

Considering only the principal values of the inverse trigonometric functions, the value of $\tan \left(\sin ^{-1}\left(\frac{3}{5}\right)-2 \cos ^{-1}\left(\frac{2}{\sqrt{5}}\right)\right)$ is

$$ \begin{array}{r}If\,\,\,\, y=\tan ^{-1}\left(\frac{1}{1+x+x^2}\right)+\tan ^{-1}\left(\frac{1}{x^2+3 x+3}\right) +\tan ^{-1}\left(\frac{1}{x^2+5 x+7}\right) \end{array} $$

then the value of $y^{\prime}(0)$ is

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

The number of positive integral solutions of $\tan ^{-1} x+\cos ^{-1}\left(\frac{y}{\sqrt{1+y^2}}\right)=\sin ^{-1}\left(\frac{3}{\sqrt{10}}\right)$ are

If $y=\tan ^{-1}\left(\sqrt{\frac{1+\sin x}{1-\sin x}}\right), 0 \leqslant x<\frac{\pi}{2}$, then $y^{\prime}\left(\frac{\pi}{6}\right)=$

If $\left(\tan ^{-1} x\right)^2+\left(\cot ^{-1} x\right)^2=\frac{5 \pi^2}{8}$, then $x^2+1=$

If $\tan ^{-1}(x+1)+\tan ^{-1} x+\tan ^{-1}(x-1)=\tan ^{-1} 3$, then for $x<0$ the value of $500 x^4+270 x^2+997=$

If $y=\tan ^{-1}\left(\frac{4 x}{1+5 x^2}\right)+\cot ^{-1}\left(\frac{3-2 x}{2+3 x}\right)$, then $\frac{\mathrm{d} y}{\mathrm{~d} x}$ is equal to

The principal value of $\cos ^{-1}\left[\frac{1}{\sqrt{2}}\left(\cos \frac{9 \pi}{10}-\sin \frac{9 \pi}{10}\right)\right]$ is

If $\left(\cos ^{-1} x\right)^2-\left(\sin ^{-1} x\right)^2>0$, then

If $0 \leqslant \cos ^{-1} x \leqslant \pi$ and $\frac{-\pi}{2} \leqslant \sin ^{-1} x \leqslant \frac{\pi}{2}$, then at $x=\frac{1}{5}$ the value of $\cos \left(2 \cos ^{-1} x+\sin ^{-1} x\right)$ is

The value of $\tan ^2\left(\sec ^{-1} 4\right)+\cot ^2\left(\operatorname{cosec}^{-1} 3\right)$ is

Considering only the principal values of the inverse trigonometric function, the value of $\tan \left(\cos ^{-1} \frac{1}{5 \sqrt{2}}-\sin ^{-1} \frac{4}{\sqrt{17}}\right)$ is

The number of solutions of $\tan ^{-1}\left(x+\frac{2}{x}\right)-\tan ^{-1}\left(\frac{4}{x}\right)-\tan ^{-1}\left(x-\frac{2}{x}\right)=0$ are

The derivative of $\tan ^{-1}\left(\frac{\sqrt{1+x^2}-1}{x}\right)$ w.r.t. $\tan ^{-1}\left(\frac{2 x \sqrt{1-x^2}}{1-2 x^2}\right)$ at $x=0$ is

If $y=\sin ^{-1}\left(\frac{2 x}{1+x^2}\right)+\sec ^{-1}\left(\frac{1+x^2}{1-x^2}\right)$ then the value of $\frac{d y}{d x}$ at $x=\sqrt{3}$ is

$$ \cot ^{-1}\left(2 \cos \left(2 \operatorname{cosec}^{-1}(\sqrt{2})\right)\right)=\ldots $$

If $3 \sin ^{-1}\left(\frac{2 x}{1+x^2}\right)-4 \cos ^{-1}\left(\frac{1-x^2}{1+x^2}\right)+2 \tan ^{-1}\left(\frac{2 x}{1-x^2}\right)=\frac{\pi}{3}$ then the value of $x=$

If $\sin ^{-1} x+\sin ^{-1} y=\frac{\pi}{3}$ and $\cot ^{-1}\left(\frac{1}{x}\right)-\cot ^{-1}\left(\frac{1}{y}\right)=0$ then $2 x^2+y^2-x y=$ $\qquad$

The value of $\sin \left[\tan ^{-1}\left(\frac{1-x^2}{2 x}\right)+\cos ^{-1}\left(\frac{1-x^2}{1+x^2}\right)\right]$ is

If $\sin ^{-1} \frac{1}{3}+\sin ^{-1} \frac{3}{5}+\sin ^{-1} x=\frac{\pi}{2}$ then $x=$

The value of $2 \tan ^{-1} \frac{1}{2}+\tan ^{-1} \frac{3}{8}$ is

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

If $\tan ^{-1}\left(\frac{x}{2}\right)+\tan ^{-1}\left(\frac{y}{2}\right)+\tan ^{-1}\left(\frac{z}{2}\right)=\frac{\pi}{2} \quad$ then $x y+y z+z x=$

$$ \begin{aligned} &\text { The value of }\\ &\begin{aligned} \sin ^{-1}\left(-\frac{1}{\sqrt{2}}\right)+\cos ^{-1} & \left(-\frac{1}{2}\right) -\cot ^{-1}\left(-\frac{1}{\sqrt{3}}\right)+\tan ^{-1}(-\sqrt{3}) \text { is } \end{aligned} \end{aligned} $$

If $a^2+b^2+c^2=r^2$, then the value of $\tan ^{-1}\left(\frac{\mathrm{ab}}{\mathrm{cr}}\right)+\tan ^{-1}\left(\frac{\mathrm{bc}}{\mathrm{ar}}\right)+\tan ^{-1}\left(\frac{\mathrm{ca}}{\mathrm{br}}\right)=$

If $\sin \left(\sin ^{-1} \frac{1}{5}+\cos ^{-1} x\right)=1$, then the value of $x$ is

If $\tan ^{-1}\left(\frac{1}{4}\right)+\tan ^{-1}\left(\frac{2}{9}\right)=\frac{1}{2} \cos ^{-1} x$, then $x$ is

If $\cos ^{-1}\left(\frac{12}{13}\right)+\sin ^{-1}\left(\frac{3}{5}\right)=\sin ^{-1} \mathrm{P}$, then the value of $P$ is

If $y=\tan ^{-1}\left(\frac{2+3 x}{3-2 x}\right)+\tan ^{-1}\left(\frac{4 x}{1+5 x^2}\right)$, then $\frac{d y}{d x}=$

The domain of the function $\mathrm{f}(x)=\frac{\sin ^{-1}(x-3)}{\sqrt{9-x^2}}$ is

If $\tan ^{-1}\left(\frac{x+1}{x-1}\right)+\tan ^{-1}\left(\frac{x-1}{x}\right)=\tan ^{-1}(-7)$, then $x$ is equal to

The value of $\tan ^{-1}(-\sqrt{3})-\sin ^{-1}\left(\frac{1}{\sqrt{2}}\right)+\cos ^{-1}\left(\frac{-1}{2}\right)$ is

If $\sin ^{-1}\left(\frac{x}{5}\right)+\operatorname{cosec}^{-1}\left(\frac{5}{4}\right)=\frac{\pi}{2}$, then the value of $x$ is

Let the function $g:(-\infty, \infty) \rightarrow\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$ be given by $g(u)=2 \tan ^{-1}\left(e^u\right)-\frac{\pi}{2}$. Then $g$ is

If $\sin ^{-1}\left(\frac{x}{13}\right)+\operatorname{cosec}^{-1}\left(\frac{13}{12}\right)=\frac{\pi}{2}$, then the value of

If $\cos ^{-1} x=\alpha(0< x < 1)$ and $\sin ^{-1}\left(2 x \sqrt{1-x^2}\right)+\sec ^{-1}\left(\frac{1}{2 x^2-1}\right)=\frac{2 \pi}{3}$, then $\alpha$ is

If $\cot ^{-1}(\sqrt{\cos \alpha})-\tan ^{-1}(\sqrt{\cos \alpha})=x$, then the value of $\sin x$ is

If $0< x<1$, then $\sqrt{1+x^2}\left[\left\{x \cos \left(\cot ^{-1} x\right)+\sin \left(\cot ^{-1} x\right)\right\}^2-1\right]^{\frac{1}{2}}$ is equal to

$2 \pi-\left(\sin ^{-1} \frac{4}{5}+\sin ^{-1} \frac{5}{13}+\sin ^{-1} \frac{16}{65}\right)$ is equal to

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

The value of $\tan ^{-1}\left\{\frac{\sqrt{1+x}-\sqrt{1-x}}{\sqrt{1+x}+\sqrt{1-x}}\right\}+\frac{1}{2} \cos ^{-1} x$ is

$$\cos \left[\sin ^{-1}\left(\frac{3}{5}\right)+\cos ^{-1}\left(\frac{12}{13}\right)\right]=$$

$\tan \left(\cos ^{-1} \frac{1}{\sqrt{2}}+\tan ^{-1} \frac{1}{2}\right)=$

If $y=\sin ^2\left(\cot ^{-1} \sqrt{\frac{1+x}{1-x}}\right)$, then $\frac{\mathrm{d} y}{\mathrm{~d} x}$ has the value

The value of $\cos \left(2 \cos ^{-1} x+\sin ^{-1} x\right)$ at $x=\frac{1}{5}$ is

If $x, y, z$ are in Arithmetic Progression and $\tan ^{-1} x, \tan ^{-1} y, \tan ^{-1} z$ are also in Arithmetic progression, where $x, z>0$ and $x z<1, y<1$, then

Derivative of $\tan ^{-1} \sqrt{\frac{1-x}{1+x}}$ w.r.t. $\cos ^{-1}\left(4 x^3-3 x\right)$ is

If $\tan ^{-1}(x+2)+\tan ^{-1}(x-2)-\tan ^{-1}\left(\frac{1}{2}\right)=0$, then one value of $x$ is

The numerical value of $\tan \left(2 \tan ^{-1}\left(\frac{1}{5}\right)+\frac{\pi}{4}\right)$

The value of $\tan ^{-1}\left(\tan \frac{7 \pi}{6}\right)$ is

If $\cos ^{-1} x+\cos ^{-1} y+\cos ^{-1} z=\pi$ and $x^2+y^2+z^2+k x y z=1$, then k is

$2 \pi-\left(\sin ^{-1} \frac{4}{5}+\sin ^{-1} \frac{5}{13}+\sin ^{-1} \frac{16}{65}\right)$ is equal to

The value of $\cos ^{-1}\left\{\frac{1}{\sqrt{2}}\left(\cos \frac{9 \pi}{10}-\sin \frac{9 \pi}{10}\right)\right\}$ is

Domain of definition of the real valued function $f(x)=\sqrt{\sin ^{-1}(2 x)+\frac{\pi}{6}}$ is

$$\frac{\mathrm{d}}{\mathrm{~d} x}\left(\cos ^{-1}\left(\frac{x-\frac{1}{x}}{x+\frac{1}{x}}\right)\right)=$$

The approximate value of $\tan ^{-1}(0.999)$ is (use $\pi=3.1415$ )

The value of $\frac{\tan ^{-1}(\sqrt{3})-\sec ^{-1}(-2)}{\operatorname{cosec}^{-1}(-\sqrt{2})+\cos ^{-1}\left(\frac{-1}{2}\right)}$

If $\cot ^{-1}(7)+\cot ^{-1}(8)+\cot ^{-1}(18)=\cot ^{-1} x$, then the value of $x$ is

If $\cos ^{-1} x+\cos ^{-1} y+\cos ^{-1} z=3 \pi$, then the value of $x^2+y^2+z^2-2 x y z$ is

If $\mathrm{f}(x)=\cos ^{-1} x, \mathrm{~g}(x)=\mathrm{e}^x$ and $\mathrm{h}(x)=\mathrm{g}(\mathrm{f}(x))$, then $\frac{\mathrm{h}^{\prime}(x)}{\mathrm{h}(x)}=$

The value of $\cot \left(\operatorname{cosec}^{-1} \frac{5}{3}+\tan ^{-1} \frac{2}{3}\right)$ is

If $\sin \left(\cot ^{-1}(x+1)\right)=\cos \left(\tan ^{-1} x\right)$ then considering positive square roots, $x$ has the value ___________

Considering only the Principal values of inverse functions, the set

$$A=\left\{x \geq 0 \left\lvert\, \tan ^{-1}(2 x)+\tan ^{-1}(3 x)=\frac{\pi}{4}\right.\right\}$$

If $0< x < 1$, then

$$\sqrt{1+x^2}\left[\left\{x \cos \left(\cot ^{-1} x\right)+\sin \left(\cot ^{-1} x\right)\right\}^2-1\right]^{\frac{1}{2}}=$$

If $y=\tan ^{-1}\left(\frac{3+2 x}{2-3 x}\right)+\tan ^{-1}\left(\frac{3 x}{1+4 x^2}\right)$, then $\frac{\mathrm{d} y}{\mathrm{~d} x}$ is equal to

The number of real solutions of $\tan ^{-1} \sqrt{x(x+1)}+\sin ^{-1} \sqrt{x^2+x+1}=\frac{\pi}{2}$ is

$\tan \left(\frac{\pi}{4}+\frac{1}{2} \cos ^{-1}\left(\frac{\mathrm{a}}{\mathrm{b}}\right)\right)+\tan \left(\frac{\pi}{4}-\frac{1}{2} \cos ^{-1}\left(\frac{\mathrm{a}}{\mathrm{b}}\right)\right)$ is

The value of $\tan \left(2 \tan ^{-1}\left(\frac{\sqrt{5}-1}{2}\right)\right)$ is

Let $f(\theta)=\sin \left(\tan ^{-1}\left(\frac{\sin \theta}{\sqrt{\cos 2 \theta}}\right)\right)$, where $\frac{-\pi}{4}<\theta<\frac{\pi}{4}$, then the value of $\frac{d}{d(\tan \theta)}(f(\theta))$ is

The value of $\cos \left(2 \cos ^{-1} x+\sin ^{-1} x\right)$ at $x=\frac{1}{5}$ where $0 \leq \cos ^{-1} x \leq \pi$ and $-\frac{\pi}{2} \leq \sin ^{-1} x \leq \frac{\pi}{2}$, is

Considering only the principal values of inverse function, the set

$$A=\left\{x \geq 0 / \tan ^{-1}(2 x)+\tan ^{-1}(3 x)=\frac{\pi}{4}\right\}$$

The number of real solutions of

$\tan ^{-1} \sqrt{x(x+1)}+\sin ^{-1} \sqrt{x^2+x+1}=\frac{\pi}{2}$ is

The value of $\sin \left(2 \cos ^{-1}\left(-\frac{3}{5}\right)\right)$ is

If $$\sum_\limits{r=1}^{50} \tan ^{-1} \frac{1}{2 r^2}=p$$ then $$\tan p$$ is

If $$\cos ^{-1} x-\cos ^{-1} \frac{y}{3}=\alpha$$, where $$-1 \leq x \leq 1$, $-3 \leq y \leq 3, x \leq \frac{y}{3}$$, then for all $$x, y, 9 x^2-6 x y \cos \alpha+y^2$$ is equal to

The value of $$\tan ^{-1}\left(\frac{1}{8}\right)+\tan ^{-1}\left(\frac{1}{2}\right)+\tan ^{-1}\left(\frac{1}{5}\right)$$ is

Given $$0 \leq x \leq \frac{1}{2}$$, then the value of $$\tan \left(\sin ^{-1}\left(\frac{x}{\sqrt{2}}+\frac{\sqrt{1-x^2}}{\sqrt{2}}\right)-\sin ^{-1} x\right)$$ is

If $$\cos ^{-1} x+\cos ^{-1} y+\cos ^{-1} z=3 \pi$$, then the value of $$x^{2025}+x^{2026}+x^{2027}$$ is

The value of $$\tan \left(\sin ^{-1}\left(\frac{3}{5}\right)+\tan ^{-1}\left(\frac{2}{3}\right)\right)$$ is

If $$\left(\tan ^{-1} x\right)^2+\left(\cot ^{-1} x\right)^2=\frac{5 \pi^2}{8}$$, then the value of $$x$$ is

The principal value of $$\sin ^{-1}(\sin (3 \pi / 4))$$ is

$$x, y, z$$ are in G.P. and $$\tan ^{-1} x, \tan ^{-1} y, \tan ^{-1} z$$ are in A.P., then

The value of $$x$$, for which $$\sin \left(\cot ^{-1}(x)\right)=\cos \left(\tan ^{-1}(1+x)\right)$$, is

If $$\tan ^{-1}\left(\frac{1-x}{1+x}\right)=\frac{1}{2} \tan ^{-1} x$$, then $$x$$ is

If $$x=\operatorname{cosec}\left(\tan ^{-1}\left(\cos \left(\cot ^{-1}\left(\sec \left(\sin ^{-1} a\right)\right)\right)\right)\right), \mathrm{a} \in[0,1]$$

The value of $$\sin \left(\cot ^{-1} x\right)$$ is

If $$\cos ^{-1} \sqrt{\mathrm{p}}+\cos ^{-1} \sqrt{1-\mathrm{p}}+\cos ^{-1} \sqrt{1-\mathrm{q}}=\frac{3 \pi}{4}$$, then $$\mathrm{q}$$ is

$$\pi+\left(\sin ^{-1} \frac{4}{5}+\sin ^{-1} \frac{5}{13}+\sin ^{-1} \frac{16}{65}\right)$$ is equal to

If $$\alpha=3 \sin ^{-1} \frac{6}{11}$$ and $$\beta=3 \cos ^{-1}\left(\frac{4}{9}\right)$$, where the inverse trigonometric functions take only the principal values, then the correct option is

$$\text { The value of } \sec ^2\left(\tan ^{-1} 2\right)+\operatorname{cosec}^2\left(\cot ^{-1} 3\right) \text { is }$$

The value of $$2 \tan ^{-1} \frac{1}{2}+\tan ^{-1} \frac{1}{7}$$

$$\text { If } y=\sqrt{\frac{1-\sin ^{-1}(x)}{1+\sin ^{-1}(x)}} \text {, then } \frac{\mathrm{d} y}{\mathrm{~d} x} \text { at } x=0 \text { and } y=1 \text { is }$$

The domain of the function $$\mathrm{f}(x)=\sin ^{-1}\left(\frac{|x|+5}{x^2+1}\right)$$ is $$(-\infty,-a] \cup[a, \infty)$$. Then a is equal to

The value of $$\tan ^{-1}(1)+\cos ^{-1}\left(-\frac{1}{2}\right)+\sin ^{-1}\left(-\frac{1}{2}\right)$$ is

If $$\tan ^{-1} a+\tan ^{-1} b+\tan ^{-1} c=\pi$$, then which of the following statement is true?

Considering only the principal values of an inverse function, the set

$$\mathrm{A}=\left\{x \geq 0 / \tan ^{-1} x+\tan ^{-1} 6 x=\frac{\pi}{4}\right\}$$

The solution of the equation $$\tan ^{-1}(1+x)+\tan ^{-1}(1-x)=\frac{\pi}{2}$$ is

The value of $$\tan ^{-1}\left(\frac{\sqrt{1+x^2}+\sqrt{1-x^2}}{\sqrt{1+x^2}-\sqrt{1-x^2}}\right), |x| < \frac{1}{2}, x \neq 0$$

If $$\sin ^{-1} x+\cos ^{-1} y=\frac{3 \pi}{10}$$, then the value of $$\cos ^{-1} x+\sin ^{-1} y$$ is

The value of $$\cot \left(\sum_\limits{n=1}^{23} \cot ^{-1}\left(1+\sum_\limits{k=1}^n 2 k\right)\right)$$ is

If $$\cos ^{-1} x-\cos ^{-1} \frac{y}{3}=\alpha$$, where $$-1 \leq x \leq 1, -3 \leq y \leq 3, x \leq \frac{y}{3}$$, then for all $$x, y$$ $$9 x^2-6 x y \cos \alpha+y^2$$ is equal to

If $$\tan ^{-1}\left(\frac{1-x}{1+x}\right)=\frac{1}{2} \tan ^{-1} x$$, then $$x$$ has the value

The value of $$\sin \left(2 \sin ^{-1} 0.8\right)$$ is equal to

The principal value of $$\sin ^{-1}\left(\sin \left(\frac{2 \pi}{3}\right)\right)$$ is

The value of $$\tan ^{-1} 2+\tan ^{-1} 3$$ is

If $$y=\tan ^{-1}\left[\frac{\log \left(\frac{e}{x^2}\right)}{\log \left(e x^2\right)}\right]+\tan ^{-1}\left[\frac{3+2 \log x}{1-6 \log x}\right]$$, then $$\frac{d^2 y}{d x^2}=$$

The value of $$\sin ^{-1}\left(\frac{-1}{2}\right)+\sin ^{-1}\left(\frac{-\sqrt{3}}{2}\right)$$ is,

If $$\tan ^{-1}(2 x)+\tan ^{-1}(3 x)=\frac{\pi}{4}$$, where $$x>0$$, then $$x=$$

$$\tan \left(\cos ^{-1}\left(\frac{4}{5}\right)+\tan ^{-1}\left(\frac{2}{3}\right)\right)=$$

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

$$\tan ^{-1}\left(\tan \frac{5 \pi}{6}\right)+\cos ^{-1}\left(\cos \frac{13 \pi}{6}\right)=$$

$$y=\tan ^{-1}\left(\sqrt{\frac{1+\sin x}{1-\sin x}}\right), 0 \leq x < \frac{\pi}{2}$$, then $$\frac{d y}{d x}$$ at $$x=\frac{\pi}{6}$$ is

If $$y=\tan ^{-1}\left[\frac{1}{1+x+x^2}\right]+\tan ^{-1}\left[\frac{1}{x^2+3 x+3}\right], x>0$$, then $$\frac{d y}{d x}=$$

If $$\sin ^{-1}\left(\frac{3}{5}\right)+\cos ^{-1}\left(\frac{12}{13}\right)=\sin ^{-1} \alpha$$, then $$\alpha=$$

$$\tan ^{-1}\left(\frac{x-1}{x-2}\right)+\tan ^{-1}\left(\frac{x+1}{x+2}\right)=\frac{\pi}{4}$$, then the values of $$x$$ are

$$\sin ^{-1}\left[\sin \left(-600^{\circ}\right)\right]+\cot ^{-1}(-\sqrt{3})=$$

$$\cos ^{-1}\left(\frac{4}{5}\right)+\cos ^{-1}\left(\frac{12}{13}\right)=$$

If $$4 \sin ^{-1} x+6 \cos ^{-1} x=3 \pi$$, where $$-1 \leq x \leq 1$$, then $$x=$$

$$\left[\sin \left(\tan ^{-1} \frac{3}{4}\right)\right]^2+\left[\sin \left(\tan ^{-1} \frac{4}{3}\right)\right]^2=$$

The value of $$\tan ^{-1}\left(\frac{1}{3}\right)+\tan ^{-1}\left(\frac{1}{5}\right)+\tan ^{-1}\left(\frac{1}{7}\right)+\tan ^{-1}\left(\frac{1}{8}\right)$$ is

The value of $$\sin ^{-1}\left(-\frac{1}{2}\right)+\cos ^{-1}\left(-\frac{\sqrt{3}}{2}\right)$$ is

Derivative of $\sin ^{-1}\left(\frac{t}{\sqrt{1+t^2}}\right)$ with respect to $\cos ^{-1}\left(\frac{1}{\sqrt{1+t^2}}\right)$ is

$\sin \left[3 \sin ^{-1}(0.4)\right]=\ldots \ldots$

If $4 \sin ^{-1} x+6 \cos ^{-1} x=3 \pi$ then $x=$ ............

If $f(x)=\cos ^{-1}\left[\frac{1-(\log x)^2}{1+(\log x)^2}\right]$, then $f^{\prime}(e)=\ldots$

The value of $\tan ^{-1} \frac{1}{3}+\tan ^{-1} \frac{1}{5}+\tan ^{-1} \frac{1}{7}+\tan ^{-1} \frac{1}{8}$ is ...........