CBSE CLASS 12 INVERSE TRIGONOMETRIC FUNCTIONS IMPORTANT QUESTIONS PDF

INVERSE TRIGONOMETRIC FUNCTIONS CLASS 12 IMPORTANT QUESTIONS

Inverse Trigonometric Functions Class 12 Important Questions PDF

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1. Evaluate the following :
$\begin{array}{l}{\text{i) sin}}^{-1}\left(\sin \frac{\pi }{4}\right){\text{ ii) cos}}^{-1}\left(\cos \frac{2\pi }{3}\right){\text{ iii) tan}}^{-1}\left(\tan \frac{\pi }{3}\right)\\ {\text{iv) sin}}^{-1}\left(\sin \frac{2\pi }{3}\right){\text{ v) cos}}^{-1}\left(\cos \frac{7\pi }{6}\right){\text{ vi) tan}}^{-1}\left(\tan \frac{2\pi }{3}\right)\\ \text{vii) cos}\left({\cos }^{-1}\left(-\frac{\sqrt{3}}{2}\right)+\frac{\pi }{6}\right){\text{ viii) sin}}^{-1}\left(\sin \left(-{600}^o\right)\right)\\ {\text{ix)cos}}^{-1}\left(\cos \left(-{680}^0\right)\right)\end{array}$

2. Express in the simplest form:
$\begin{array}{l}\text{i)}{\text{tan}}^{-1}\left(\frac{\cos x}{1-\sin x}\right),-\frac{\pi }{2}<x<\frac{\pi }{2}\\ {\text{ii) tan}}^{-1}\left(\frac{\cos x-\sin x}{\cos x+\sin x}\right),-\frac{\pi }{4}<x<\frac{\pi }{4}\end{array}$

3. Prove that :
$\begin{array}{l}\text{i)}{\text{cot}}^{-1}\left\{\frac{\sqrt{1+\sin x}+\sqrt{1-\sin x}}{\sqrt{1+\sin x}-\sqrt{1-\sin x}}\right\}=\frac{x}{2},0<x<\frac{\pi }{2}\\ \text{ii)}{\text{cot}}^{-1}\left\{\frac{\sqrt{1+\sin x}+\sqrt{1-\sin x}}{\sqrt{1+\sin x}-\sqrt{1-\sin x}}\right\}=\frac{\pi }{2}-\frac{x}{2},\frac{\pi }{2}<x<\pi \\ {\text{iii) tan}}^{-1}\left\{\frac{\sqrt{1+x}-\sqrt{1-x}}{\sqrt{1+x}+\sqrt{1-x}}\right\}=\frac{\pi }{4}-\frac{1}{2}{\cos }^{-1}x,0<x<1\end{array}$

4.Prove that :
${\tan }^{-1}\left\{\frac{\sqrt{1+x^2}+\sqrt{1-x^2}}{\sqrt{1+x^2}-\sqrt{1-x^2}}\right\}=\frac{\pi }{4}+\frac{1}{2}{\cos }^{-1}{x}^2,-1<x<1$

5. Simplify each of the following :
$\begin{array}{l}\text{i) }{\cos }^{-1}\left(\frac{3}{5}\cos x+\frac{4}{5}\sin x\right),\text{where -}\frac{3\pi }{4}\le x\le \frac{\pi }{4}\\ {\text{ii) sin}}^{-1}\left(\frac{5}{13}\cos x+\frac{12}{13}\sin x\right)\end{array}$

6. Simplify each of the following :
$\begin{array}{l}{\text{i) sin}}^{-1}\left(\frac{\sin x+\cos x}{\sqrt{2}}\right),-\frac{\pi }{4}<x<\frac{\pi }{4}\\ {\text{ii) cos}}^{-1}\left(\frac{\sin x+\cos x}{\sqrt{2}}\right),\frac{\pi }{4}<x<\frac{5\pi }{4}\\ {\text{iii)sin}}^{-1}\left(\frac{\sin x+\cos x}{\sqrt{2}}\right),\frac{\pi }{4}<x<\frac{5\pi }{4}\\ {\text{iv)cos}}^{-1}\left(\frac{\sin x+\cos x}{\sqrt{2}}\right),\frac{5\pi }{4}<x<\frac{9\pi }{4}\end{array}$

7. Evaluate :
$\begin{array}{l}\text{i) sin}\left({\cot }^{-1}x\right)\\ \text{ii) cos}\left({\tan }^{-1}x\right)\\ \text{iii) cos}\left({\sin }^{-1}\frac{1}{4}+{\sec }^{-1}\frac{4}{3}\right)\\ \text{iv) sin}\left[{\cot }^{-1}\left\{\cos \left({\tan }^{-1}x\right)\right\}\right]\end{array}$

8. Prove that :
$\begin{array}{l}{\text{i) tan}}^2\left({\sec }^{-1}2\right)+{\cot }^2\left(\cos e{c}^{-1}3\right)=11\\ \text{ii)cos}\left[{\tan }^{-1}\left\{\sin \left({\cot }^{-1}x\right)\right\}\right]=\sqrt{\frac{x^2+1}{x^2+2}}\\ \text{iii)cos}\left({\sin }^{-1}\frac{3}{5}+{\cot }^{-1}\frac{3}{2}\right)=\frac{6}{5\sqrt{13}}\end{array}$

9. Find the value of
$\cot \left({\tan }^{-1}a+{\cot }^{-1}a\right)$

10. If $-1\le x,y\le 1{\text{ such that sin}}^{-1}x+{\sin }^{-1}y=\frac{\pi }{2},{\text{ find the value of cos}}^{-1}x+{\cos }^{-1}y$

11. If ${\tan }^{-1}x-{\cot }^{-1}x={\tan }^{-1}\frac{1}{\sqrt{3}},\text{ find the value of }x.$

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

13. Solve : $\sin \left\{{\sin }^{-1}\frac{1}{5}+{\cos }^{-1}x\right\}=1$

14. Prove that : ${\tan }^{-1}1+{\tan }^{-1}2+{\tan }^{-1}3=\pi$

15.Prove that : ${\sin }^{-1}\frac{12}{13}+{\cos }^{-1}\frac{4}{5}+{\tan }^{-1}\frac{63}{16}=\pi$ ${\sin }^{-1}\frac{12}{13}+{\cos }^{-1}\frac{4}{5}+{\tan }^{-1}\frac{63}{16}=\pi$

16. Prove that : ${\tan }^{-1}\frac{1}{2}+{\tan }^{-1}\frac{1}{5}+{\tan }^{-1}\frac{1}{8}=\frac{\pi }{4}$

17. Prove that : ${\tan }^{-1}\frac{1}{5}+{\tan }^{-1}\frac{1}{7}+{\tan }^{-1}\frac{1}{3}+{\tan }^{-1}\frac{1}{8}=\frac{\pi }{4}$
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