Chapter 6

Prove that \(\sin ^{-1}\left(\cos \left(\sin ^{-1} x\right)\right)+\cos ^{-1}\left(\sin \left(\cos ^{-1} x\right)\right)\)

The set of values of \(k\) for which \(x^{2}-k x+\sin ^{-1}\) \((\sin 4)>0\) for all real \(x\) is (a) \(\\{0\\}\) (b) \((-2,2)\) (c) \(\bar{R}\) (d) None of these.

Prove that \(\tan ^{-1}\left\\{\operatorname{cosec}\left(\tan ^{-1} x\right)-\tan \left(\cot ^{-1} x\right)\right\\}=\frac{1}{2} \tan ^{-1} x\) where \(x \neq 0\)

If \(x<0\) then value of \(\tan ^{-1}(x)+\tan ^{-1}\left(\frac{1}{x}\right)=\) (a) \(\frac{\pi}{2}\) (b) \(-\frac{\pi}{2}\) (c) 0 (d) None of these.

Prove that \(\tan \left(\tan ^{-1} x+\tan ^{-1} y+\tan ^{-1} z\right)\) \(=\cot \left(\cot ^{-1} x+\cot ^{-1} y+\cot ^{-1} z\right)\)

If \(\sin ^{-1} x+\sin ^{-1} y=\frac{2 \pi}{3}\), then \(\cos ^{-1} x+\cos ^{-1} y\) is (a) \(\frac{2 \pi}{3}\) (b) \(\frac{\pi}{3}\) (c) \(\frac{\pi}{6}\) (d) \(\pi\)

Prove that \(\sin \left(\cot ^{-1}\left(\tan \left(\cos ^{-1} x\right)\right)\right)=x \quad \forall x \in(0,1]\)

Let \(f(x)=\sin ^{-1} x+\cos ^{-1} x\). Then \(\frac{\pi}{2}\) is equal to (a) \(f\left(\frac{1}{2}\right)\) (b) \(f\left(k^{2}-2 k+3\right), k \varepsilon R\) (c) \(f\left(\frac{1}{1+k^{2}}\right), k \varepsilon R\) (d) \(f(-2)\)

Prove that \(\sin \left(\operatorname{cosec}^{-1}\left(\cot \left(\tan ^{-1} x\right)\right)\right)=x\) \(\forall x \in(0,1]\)

Which one of the following is correct? (a) \(\tan 1>\tan ^{-1} 1\) (b) \(\tan 1<\tan ^{-1} 1\) (c) \(\tan 1=\tan ^{-1} 1\) (d) None.

Find the value of \(\sin ^{-1}(\sin 5)+\cos ^{-1}(\cos 10)+\tan ^{-1}(\tan (-6))\) \(+\cot ^{-1}(\cot (-10))\)

If \(a \sin ^{-1} x-b \cos ^{-1} x=c\), then the value of a \(\sin ^{-1} x+b \cos ^{-1} x\) is (a) 0 (b) \(\frac{\pi a b+c(b-a)}{a+b}\) (c) \(\frac{\pi a b-c(b-a)}{a+b}\) (d) \(\frac{\pi}{2}\)

Find the simplest value of \(\cos ^{-1} x+\cos ^{-1}\left(\frac{x}{2}+\frac{\sqrt{3-3 x^{2}}}{2}\right), \forall x \in\left(\frac{1}{2}, 1\right)\)

The number of solutions of the equation \(\sin ^{-1}(1-x)-2 \sin ^{-1} x=\frac{\pi}{2}\) (a) 0 (b) 1 (c) 2 (d) More than two

Find the value of \(\tan ^{-1}\left(\frac{1}{\sqrt{2}}\right)-\tan ^{-1}\left(\frac{\sqrt{5-2 \sqrt{6}}}{1+\sqrt{6}}\right)\)

The smallest and the largest values of \(\tan ^{-1}\left(\frac{1-x}{1+x}\right), 0 \leq x \leq 1\) are (a) \(0, \pi\) (b) \(0, \frac{\pi}{4}\) (c) \(-\frac{\pi}{4}, \frac{\pi}{4}\) (d) \(\frac{\pi}{4}, \frac{\pi}{2}\)

Let \(m=\sin ^{-1}\left(a^{6}+1\right)+\cos ^{-1}\left(a^{4}+1\right)-\tan ^{-1}\left(a^{2}+1\right)\), then find the image of the line \(x+y=m\) about the \(y\)-axis.

The equation \(\sin ^{-1} x-\cos ^{-1} x=\cos ^{-1}\left(\frac{\sqrt{3}}{2}\right)\) has (a) No solution (b) Unique Solution (c) Infinite No of soln (d) None

If \(-\pi \leq x \leq 2 \pi\), then \(\cos ^{-1}(\cos x)\) is (a) \(x\) (b) \(\pi-x\) (c) \(2 \pi+x\) (d) \(2 \pi-x\)

Let \(S=\sum_{r=1}^{n} \cot ^{-1}\left(2^{r+1}+\frac{1}{2^{r}}\right)\), then find \(\lim _{n \rightarrow \infty}(S)\).

If \(\sin ^{-1} x+\cot ^{-1}\left(\frac{1}{2}\right)=\frac{\pi}{2}\), then \(x\) is equal to (a) 0 (b) \(\frac{1}{\sqrt{5}}\) (c) \(\frac{2}{\sqrt{5}}\) (d) \(\frac{\sqrt{3}}{2}\)

If \(\cos \left[\tan ^{-1}\left\\{\sin \left(\cot ^{-1} \sqrt{3}\right)\right\\}\right]=y\), then the value of \(y\) is (a) \(y=\frac{4}{5}\) (b) \(y=\frac{2}{\sqrt{5}}\) (c) \(y=-\frac{2}{\sqrt{5}}\) (d) \(y=\frac{\sqrt{3}}{2}\)

Find the number of solution of the equation \(2 \sin ^{-1}\left(\frac{2 x}{1+x^{2}}\right)=\pi x^{3}\)

If \(x=\frac{1}{5}\), then the value of \(\cos \left(\cos ^{-1} x+2 \sin ^{-1} x\right)\) is (a) \(\sqrt{\frac{24}{25}}\) (b) \(-\sqrt{\frac{24}{25}}\) (c) \(\frac{1}{5}\) (d) \(-\frac{1}{5}\)

If \(\cos ^{-1}\left(\frac{x}{a}\right)+\cos ^{-1}\left(\frac{y}{b}\right)=\alpha\), then prove that, \(\frac{x^{2}}{a^{2}}-\frac{2 x y}{a b} \cos \alpha+\frac{y^{2}}{b^{2}}=\sin ^{2} \alpha .\)

\(\tan ^{-1}\left(\frac{1}{2}\right)+\tan ^{-1}\left(\frac{1}{3}\right)\) is equal to (a) \(\frac{\pi}{4}\) (b) \(\frac{\pi}{2}\) (c) \(\frac{\pi}{3}\) (d) None of these

If \(\sin ^{-1} x+\sin ^{-1} y+\sin ^{-1} z=\pi\), then prove that, \(x \sqrt{1-x^{2}}+y \sqrt{1-y^{2}}+z \sqrt{1-z^{2}}=2 x y z\).

\(\tan ^{-1} a+\tan ^{-1} b\), where \(a>0, b>0, a b>1\) is equal to (a) \(\tan ^{-1}\left(\frac{a+b}{1-a b}\right)\) (b) \(\tan ^{-1}\left(\frac{a+b}{1-a b}\right)-\pi\) (c) \(\pi+\tan ^{-1}\left(\frac{a+b}{1-a b}\right)\) (d) \(\pi-\tan ^{-1}\left(\frac{a+b}{1-a b}\right)\)

Find the greatest and least value of the function \(f(x)=\left(\sin ^{-1} x\right)^{3}+\left(\cos ^{-1} x\right)^{3}\)

A solution to the equation \(\tan ^{-1}(1+x)+\tan ^{-1}(1-x)=\frac{\pi}{2}\) is (a) \(x=1\) (b) \(x=-1\) (c) \(x=0\) (d) \(x=\pi\)

Solve for \(x: \sin ^{-1} x+\sin ^{-1} 2 x=\frac{\pi}{2}\).

All possible values of \(p\) and \(q\) for which \(\cos ^{-1}(\sqrt{p})+\cos ^{-1}(\sqrt{1-p})+\cos ^{-1}(\sqrt{1-q})=\frac{3 \pi}{4}\) holds, is (a) \(p=1, q=1 / 2\) (b) \(q>1, p=1 / 2\) (c) \(0

Solve for \(x\) : \(\tan ^{-1}\left(\frac{1}{1+2 x}\right)+\tan ^{-1}\left(\frac{1}{1+4 x}\right)=\tan ^{-1}\left(\frac{2}{x^{2}}\right)\)

\(\tan \left(\frac{\pi}{4}+\frac{1}{2} \cos ^{-1} x\right)+\tan \left(\frac{\pi}{4}-\frac{1}{2} \cos ^{-1} x\right)\) \(x \neq 0\), is equal to (a) \(x\) (b) \(2 x\) (c) \(\frac{2}{x}\) (d) \(\frac{x}{2}\)

Solve for \(x\) : \(\tan ^{-1}(x-1)+\tan ^{-1}(x)+\tan ^{-1}(x+1)\) \(=\tan ^{-1}(3 x)\)

The value of \(\cot ^{-1}(3)+\operatorname{cosec}^{-1}(\sqrt{5})\) is (a) \(\frac{\pi}{2}\) (b) \(\frac{\pi}{3}\) (c) \(\frac{\pi}{4}\) (d) \(\frac{\pi}{6}\)

Solve for \(x: \sin ^{-1}\left(\frac{1}{\sqrt{5}}\right)+\cos ^{-1} x=\frac{\pi}{4}\).

If \(\sum_{i=1}^{2 n} \sin ^{-1} x_{i}=n \pi\), then \(\sum_{i=1}^{2 n} x_{j}\) is (a) \(n\) (b) \(2 n\) (c) \(\frac{n(n+1)}{2}\) (d) \(\frac{n(n-1)}{2}\)

Solve for \(x: \cos ^{-1}\left(\frac{x^{2}-1}{x^{2}+1}\right)+\tan ^{-1}\left(\frac{2 x}{x^{2}-1}\right)=\frac{2 \pi}{3}\).

If \(u=\cot ^{-1}(\sqrt{\tan \alpha})-\tan ^{-1}(\sqrt{\tan \alpha})\), then \(\tan \left(\frac{\pi}{4}-\frac{u}{2}\right)\) is equal to (a) \(\sqrt{\tan \alpha}\) (b) \(\sqrt{\cot \alpha}\) (c) \(\tan \alpha\) (d) \(\cot \alpha\)

Solve for \(x\) : \(2 \tan ^{-1} x=\cos ^{-1}\left(\frac{1-a^{2}}{1+a^{2}}\right)-\cos ^{-1}\left(\frac{1-b^{2}}{1+b^{2}}\right)\), \(a>0, b>0 .\)

The value of \(\tan ^{-1}\left(\frac{a}{b+c}\right)+\tan ^{-1}\left(\frac{b}{a+c}\right)\), if \(\angle C=90\), in triangle \(A B C\) is (a) \(\frac{\pi}{4}\) (b) \(\frac{\pi}{3}\) (c) \(\frac{\pi}{2}\) (d) \(\pi\)

Solve for \(x\) : \(\cot ^{-1} x+\cot ^{-1}\left(n^{2}-x+1\right)=\cot ^{-1}(n-1) .\)

If \(\cot ^{-1}\left(\frac{n}{\pi}\right)>\frac{\pi}{6}, n \varepsilon N\), then the maximum value of \(" n\) 'is (a) 1 (b) 5 (c) 9 (d) None of these

Solve for \(x\) : \(\tan ^{-1}\left(\frac{x-1}{x+1}\right)+\tan ^{-1}\left(\frac{2 x-1}{2 x+1}\right)=\tan ^{-1}\left(\frac{23}{36}\right)\)

\(\sin ^{-1} x>\cos ^{-1} x\) holds for (a) all values of \(x\) (b) \(x \varepsilon\left(0, \frac{1}{\sqrt{2}}\right)\) (c) \(\left(\frac{1}{\sqrt{2}}, 1\right)\) (d) \(x=0.75\)

Solve for \(x\) : \(\sec ^{-1}\left(\frac{x}{a}\right)-\sec ^{-1}\left(\frac{x}{b}\right)=\sec ^{-1} b-\sec ^{-1} a .\)

The value of \(\cos \left(\frac{1}{2} \cos ^{-1}\left(\frac{1}{8}\right)\right)\) is equal to (a) \(\frac{3}{4}\) (b) \(-\frac{3}{4}\)

Find the sum of \(\sum_{n=1}^{\infty} \tan ^{-1}\left(\frac{8 n}{n^{4}-2 n^{2}+5}\right)\)

The values of \(x\) satisfying \(\tan \left(\sec ^{-1} x\right)=\sin \left(\cos ^{-1}\left(\frac{1}{\sqrt{5}}\right)\right)\) is (a) \(\pm \frac{\sqrt{5}}{3}\) (b) \(\pm \frac{3}{\sqrt{5}}\) (c) \(\pm \frac{\sqrt{3}}{5}\) (d) \(\pm \frac{3}{5}\)