Inverse Trigonometric Functions

Q38 The principal value of ${\cos ^{ - 1}}\left( { - \frac{1}{2}} \right)$ is ……….. FillBlank Q39 The value of ${\sin ^{ - 1}}\left( {\sin \frac{{3\pi }}{5}} \right)$ is ……………. FillBlank Q40 If $\cos \left( {{{\tan }^{ - 1}}x + {{\cot }^{ - 1}}\sqrt 3 } \right) = 0$, then the value of $x$ is FillBlank Q41 The set of values of ${\sec ^{ - 1}}\frac{1}{2}$ is FillBlank Q42 The principal value of ${\tan ^{ - 1}}\sqrt 3$ is……………….. FillBlank Q43 The value of ${\cos ^{ - 1}}\left( {\cos \frac{{14\pi }}{3}} \right)$ is………… FillBlank Q44 The value of $\cos \left( {{{\sin }^{ - 1}}x + {{\cos }^{ - 1}}x} \right)$, where $|x| \le 1$, is………….. FillBlank Q45 The value of $\tan \left( {\frac{{{{\sin }^{ - 1}}x + {{\cos }^{ - 1}}x}}{2}} \right)$, when $x = \frac{{\sqrt 3 }}{2}$, is ………….. FillBlank Q46 If $y = 2{\tan ^{ - 1}}x + {\sin ^{ - 1}}\left( {\frac{{2x}}{{1 + {x^2}}}} \right)$, then …………….$< y <$…..………….. FillBlank Q47 The result ${\tan ^{ - 1}}x - {\tan ^{ - 1}}y = {\tan ^{ - 1}}\left( {\frac{{x - y}}{{1 + xy}}} \right)$ is true when the value of $xy$ is……………….. FillBlank Q48 The value of ${\cot ^{ - 1}}( - x)x \in R$ in terms of ${\cot ^{ - 1}}x$ is…………… FillBlank Q12 Prove that ${\tan ^{ - 1}}\left( {\frac{{\sqrt {1 + {x^2}} + \sqrt {1 - {x^2}} }}{{\sqrt {1 + {x^2}} - \sqrt {1 - {x^2}} }}} \right) = \frac{\pi }{4} + \frac{1}{2}{\cos ^{ - 1}}{x^2}$. LA Q13 Find the simplified form of
${\cos ^{ - 1}}\left( {\frac{3}{5}\cos x + \frac{4}{5}\sin x} \right)$, where $x \in \left[ {\frac{{ - 3\pi }}{4},\frac{\pi }{4}} \right]$. LA Q14 Prove that ${\sin ^{ - 1}}\frac{8}{{17}} + {\sin ^{ - 1}}\frac{3}{5} = {\sin ^{ - 1}}\frac{{77}}{{85}}$. LA Q15 Show that ${\sin ^{ - 1}}\frac{5}{{13}} + {\cos ^{ - 1}}\frac{3}{5} = {\tan ^{ - 1}}\frac{{63}}{{16}}$. LA Q16 Prove that ${\tan ^{ - 1}}\frac{1}{4} + {\tan ^{ - 1}}\frac{2}{9} = {\sin ^{ - 1}}\frac{1}{{\sqrt 5 }}$. LA Q17 Find the value of $4{\tan ^{ - 1}}\frac{1}{5} - {\tan ^{ - 1}}\frac{1}{{239}}$. LA Q18 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? LA Q19 If ${a_1},{a_2},{a_3}, \ldots ,{a_n}$ is an arithmetic progression with common difference
$d$, then evaluate the following expression.
$\tan \left[ {{{\tan }^{ - 1}}\left( {\frac{d}{{1 + {a_1}{a_2}}}} \right) + {{\tan }^{ - 1}}\left( {\frac{d}{{1 + {a_2}{a_3}}}} \right) + {{\tan }^{ - 1}}\left( {\frac{d}{{1 + {a_3}{a_4}}}} \right)} \right.$ $\left. { + \ldots + {{\tan }^{ - 1}}\left( {\frac{d}{{1 + {a_{n - 1}}{a_n}}}} \right)} \right]$

We have, ${a_1} = a,{a_2} = a + d,{a_3} = a + 2d$
and $d = {a_2} - {a_1} = {a_3} - {a_2} = {a_4} - {a_3} = \ldots = {a_n} - {a_{n - 1}}$ LA Q20 Which of the following is the principal value branch of ${\cos ^{ - 1}}x$ ? MCQ Q21 Which of the following is the principal value branch of ${{\mathop{\rm cosec}\nolimits} ^{ - 1}}x$ ? MCQ Q22 If $3{\tan ^{ - 1}}x + {\cot ^{ - 1}}x = \pi$, then $x$ equals to MCQ Q23 The value of ${\sin ^{ - 1}}\left[ {\cos \left( {\frac{{33\pi }}{5}} \right)} \right]$ is MCQ Q24 The domain of the function ${\cos ^{ - 1}}(2x - 1)$ is MCQ Q25 The domain of the function defined by $f(x) = {\sin ^{ - 1}}\sqrt {x - 1}$ is MCQ Q26 If $\cos \left( {{{\sin }^{ - 1}}\frac{2}{5} + {{\cos }^{ - 1}}x} \right) = 0$, then $x$ is equal to MCQ Q27 The value of $\sin \left[ {2{{\tan }^{ - 1}}(0.75)} \right]$ is MCQ Q28 The value of ${\cos ^{ - 1}}\left( {\cos \frac{{3\pi }}{2}} \right)$ is MCQ Q29 The value of $2{\sec ^{ - 1}}2 + {\sin ^{ - 1}}\left( {\frac{1}{2}} \right)$ is MCQ Q30 If ${\tan ^{ - 1}}x + {\tan ^{ - 1}}y = \frac{{4\pi }}{5}$, then ${\cot ^{ - 1}}x + {\cot ^{ - 1}}y$ equals to MCQ Q31 If ${\sin ^{ - 1}}\left( {\frac{{2a}}{{1 + {a^2}}}} \right) + {\cos ^{ - 1}}\left( {\frac{{1 - {a^2}}}{{1 + {a^2}}}} \right) = {\tan ^{ - 1}}\left( {\frac{{2x}}{{1 - {x^2}}}} \right)$, where $\left. {a,x \in } \right]0,1$ MCQ Q32 The value of $\cot \left[ {{{\cos }^{ - 1}}\left( {\frac{7}{{25}}} \right)} \right]$ is MCQ Q33 The value of $\tan \left( {\frac{1}{2}{{\cos }^{ - 1}}\frac{2}{{\sqrt 5 }}} \right)$ is MCQ Q34 If $|x| \le 1$, then $2{\tan ^{ - 1}}x + {\sin ^{ - 1}}\left( {\frac{{2x}}{{1 + {x^2}}}} \right)$ is equal to MCQ Q35 If ${\cos ^{ - 1}}\alpha + {\cos ^{ - 1}}\beta + {\cos ^{ - 1}}\gamma = 3\pi$, then $\alpha (\beta + \gamma ) + \beta (\gamma + \alpha ) + \gamma (\alpha + \beta )$ equals to MCQ Q36 The number of real Solutions of the equation $\sqrt {1 + \cos 2x} = \sqrt 2 {\cos ^{ - 1}}(\cos x){\mathop{\rm in}\nolimits} \left[ {\frac{\pi }{2},\pi } \right]$ is MCQ Q37 If ${\cos ^{ - 1}}x > {\sin ^{ - 1}}x$, then MCQ Q1 Find the value of ${\tan ^{ - 1}}\left( {\tan \frac{{5\pi }}{6}} \right) + {\cos ^{ - 1}}\left( {\cos \frac{{13\pi }}{6}} \right)$. SA Q2 Evaluate $\cos \left[ {{{\cos }^{ - 1}}\left( {\frac{{ - \sqrt 3 }}{2}} \right) + \frac{\pi }{6}} \right]$. SA Q3 Prove that $\cot \left( {\frac{\pi }{4} - 2{{\cot }^{ - 1}}3} \right) = 7$. SA Q4 Find the value of ${\tan ^{ - 1}}\left( { - \frac{1}{{\sqrt 3 }}} \right) + {\cot ^{ - 1}}\left( {\frac{1}{{\sqrt 3 }}} \right) + {\tan ^{ - 1}}\left[ {\sin \left( {\frac{{ - \pi }}{2}} \right)} \right]$. SA Q5 Find the value of ${\tan ^{ - 1}}\left( {\tan \frac{{2\pi }}{3}} \right)$. SA Q6 Show that $2{\tan ^{ - 1}}( - 3) = \frac{{ - \pi }}{2} + {\tan ^{ - 1}}\left( {\frac{{ - 4}}{3}} \right)$. SA Q7 Find the real Solution of
${\tan ^{ - 1}}\sqrt {x(x + 1)} + {\sin ^{ - 1}}\sqrt {{x^2} + x + 1} = \frac{\pi }{2}$. SA Q8 Find the value of $\sin \left( {2{{\tan }^{ - 1}}\frac{1}{3}} \right) + \cos \left( {{{\tan }^{ - 1}}2\sqrt 2 } \right)$. SA Q9 If $2{\tan ^{ - 1}}(\cos \theta ) = {\tan ^{ - 1}}(2{\mathop{\rm cosec}\nolimits} \theta )$, then show that $\theta = \frac{\pi }{4}$, where $n$ is any integer. SA Q10 Show that $\cos \left( {2{{\tan }^{ - 1}}\frac{1}{7}} \right) = \sin \left( {4{{\tan }^{ - 1}}\frac{1}{3}} \right)$. SA Q11 Solve the equation $\cos \left( {{{\tan }^{ - 1}}x} \right) = \sin \left( {{{\cot }^{ - 1}}\frac{3}{4}} \right)$. SA

Q1 ${\sin ^{ - 1}}\left( { - \frac{1}{2}} \right)$ SA Q2 ${\cos ^{ - 1}}\left( {\frac{{\sqrt 3 }}{2}} \right)$ SA Q3 ${\rm{cose}}{{\rm{c}}^{ - 1}}(2)$ SA Q4 ${\tan ^{ - 1}}( - \sqrt 3 )$ SA Q5 ${\cos ^{ - 1}}\left( { - \frac{1}{2}} \right)$ SA Q6 ${\tan ^{ - 1}}( - 1)$ SA Q7 ${\sec ^{ - 1}}\left( {\frac{2}{{\sqrt 3 }}} \right)$ SA Q8 ${\cot ^{ - 1}}(\sqrt 3 )$ SA Q9 ${\cos ^{ - 1}}\left( { - \frac{1}{{\sqrt 2 }}} \right)$ SA Q10 ${\rm{cose}}{{\rm{c}}^{ - 1}}( - \sqrt 2 )$ SA Q11 ${\tan ^{ - 1}}(1) + {\cos ^{ - 1}}\left( { - \frac{1}{2}} \right) + {\sin ^{ - 1}}\left( { - \frac{1}{2}} \right)$ SA Q12 ${\cos ^{ - 1}}\left( {\frac{1}{2}} \right) + 2{\sin ^{ - 1}}\left( {\frac{1}{2}} \right)$ SA Q13 . If ${\sin ^{ - 1}}x = y,$ then

(A) $0 \le y \le \pi$

(B) $- \frac{\pi }{2} \le y \le \frac{\pi }{2}$

(C) $0 < y < \pi$

(D) $- \frac{\pi }{2} < y < \frac{\pi }{2}$ SA Q14 ${\tan ^{ - 1}}\sqrt 3 - {\sec ^{ - 1}}( - 2)$ is equal to

(A) $\pi$

(B) $- \frac{\pi }{3}$

(C) $\frac{\pi }{3}$

(D) $\frac{{2\pi }}{3}$ SA

Q ${\tan ^{ - 1}}\frac{2}{{11}} + {\tan ^{ - 1}}\frac{7}{{24}} = {\tan ^{ - 1}}\frac{1}{2}$ SA Q1 $3{\sin ^{ - 1}}x = {\sin ^{ - 1}}(3x - 4{x^3}),\;x \in \left[ { - \frac{1}{2},\;\frac{1}{2}} \right]$ SA Q2 $3{\cos ^{ - 1}}x = {\cos ^{ - 1}}(4{x^3} - 3x),\;x \in \left[ {\frac{1}{2},\;1} \right]$ SA Q4 $2{\tan ^{ - 1}}\frac{1}{2} + {\tan ^{ - 1}}\frac{1}{7} = {\tan ^{ - 1}}\frac{{31}}{{17}}$ SA Q5 ${\tan ^{ - 1}}\frac{{\sqrt {1 + {x^2}} - 1}}{x},\;x \ne 0$ SA Q6 ${\tan ^{ - 1}}\frac{1}{{\sqrt {{x^2} - 1} }},\;|x|\; > 1$ SA Q7 ${\tan ^{ - 1}}\left( {\sqrt {\frac{{1 - \cos x}}{{1 + \cos x}}} } \right),\;x < \pi$ SA Q8 ${\tan ^{ - 1}}\left( {\frac{{\cos x - \sin x}}{{\cos x + \sin x}}} \right),\;0 < x < \pi$ SA Q9 ${\tan ^{ - 1}}\left( {\frac{x}{{\sqrt {{a^2} - {x^2}} }}} \right),\;|x|\; < a$ SA Q10 ${\tan ^{ - 1}}\left( {\frac{{3{a^2}x - {x^3}}}{{{a^3} - 3a{x^2}}}} \right),\;a > 0;\;\,\frac{{ - a}}{{\sqrt 3 }} \le x \le \frac{a}{{\sqrt 3 }}.$ SA Q11 ${\tan ^{ - 1}}\left[ {2\cos \left( {2{{\sin }^{ - 1}}\frac{1}{2}} \right)} \right]$ SA Q12 $\cot ({\tan ^{ - 1}}a + {\cot ^{ - 1}}a)$ SA Q13 $\tan \frac{1}{2}\left[ {{{\sin }^{ - 1}}\frac{{2x}}{{1 + {x^2}}} + {{\cos }^{ - 1}}\frac{{1 - {y^2}}}{{1 + {y^2}}}} \right],\;\,|x|\; < 1,\;y > 0$ and $xy < 1$ SA Q14 If $\sin \left( {{{\sin }^{ - 1}}\frac{1}{5} + {{\cos }^{ - 1}}x} \right) = 1,$ then find the value of x. SA Q15 If ${\tan ^{ - 1}}\frac{{x - 1}}{{x - 2}} + {\tan ^{ - 1}}\frac{{x + 1}}{{x + 2}} = \frac{\pi }{4},$ then find the value of x SA Q16 ${\sin ^{ - 1}}\left( {\sin \frac{{2\pi }}{3}} \right)$ SA Q17 ${\tan ^{ - 1}}\left( {\tan \frac{{3\pi }}{4}} \right)$ SA Q18 $\tan \left[ {\left( {{{\sin }^{ - 1}}\frac{3}{5}} \right) + {{\cot }^{ - 1}}\frac{3}{2}} \right]$ SA Q19 ${\cos ^{ - 1}}\left( {\cos \frac{{7\pi }}{6}} \right)$ is equal to

(A) $\frac{{7\pi }}{6}$

(B) $\frac{{5\pi }}{6}$

(C) $\frac{\pi }{3}$

(D) $\frac{\pi }{6}$ SA Q20 $\sin \left( {\frac{\pi }{3} - {{\sin }^{ - 1}}\left( { - \frac{1}{2}} \right)} \right)$ is equal to

(A) $\frac{1}{2}$

(B) $\frac{1}{3}$

(C) $\frac{1}{4}$

(D) 1 SA Q21 ${\tan ^{ - 1}}\sqrt 3 - {\cot ^{ - 1}}( - \sqrt 3 )$ is equal to

(A) $\pi$

(B) $- \frac{\pi }{2}$

(C) 0

(D) $2\sqrt 3$ SA

Q1 ${\cos ^{ - 1}}\left( {\cos \frac{{13\pi }}{6}} \right)$ SA Q2 ${\tan ^{ - 1}}\left( {\tan \frac{{7\pi }}{6}} \right)$ SA Q3 $2{\sin ^{ - 1}}\frac{3}{5} = {\tan ^{ - 1}}\frac{{24}}{7}$ SA Q4 ${\sin ^{ - 1}}\frac{8}{{17}} + {\sin ^{ - 1}}\frac{3}{5} = {\tan ^{ - 1}}\frac{{77}}{{36}}$ SA Q5 ${\cos ^{ - 1}}\frac{4}{5} + {\cos ^{ - 1}}\frac{{12}}{{13}} = co{s^{ - 1}}\frac{{33}}{{65}}$ SA Q6 ${\cos ^{ - 1}}\frac{{12}}{{13}} + {\sin ^{ - 1}}\frac{3}{5} = {\sin ^{ - 1}}\frac{{56}}{{65}}$ SA Q7 ${\tan ^{ - 1}}\frac{{63}}{{16}} = {\sin ^{ - 1}}\frac{5}{{13}} + {\cos ^{ - 1}}\frac{3}{5}$ SA Q8 ${\tan ^{ - 1}}\frac{1}{5} + {\tan ^{ - 1}}\frac{1}{7} + {\tan ^{ - 1}}\frac{1}{3} + {\tan ^{ - 1}}\frac{1}{8} = \frac{\pi }{4}$ SA Q9 ${\tan ^{ - 1}}\sqrt x = \frac{1}{2}{\cos ^{ - 1}}\left( {\frac{{1 - x}}{{1 + x}}} \right),\;\;x \in [0,\;1]$ SA Q10 ${\cot ^{ - 1}}\left( {\frac{{\sqrt {1 + \sin x} + \sqrt {1 - \sin x} }}{{\sqrt {1 + \sin x} - \sqrt {1 - \sin x} }}} \right) = \frac{x}{2},\;\;x \in \left( {0,\;\;\frac{\pi }{4}} \right)$ SA Q11 ${\tan ^{ - 1}}\left( {\frac{{\sqrt {1 + x} - \sqrt {1 - x} }}{{\sqrt {1 + x} + \sqrt {1 - x} }}} \right) = \frac{\pi }{4} - \frac{1}{2}{\cos ^{ - 1}}x,\;\; - \frac{1}{{\sqrt 2 }} \le x \le 1$

[Hint : Put $x = \cos 2\theta ]$ SA Q12 $\frac{{9\pi }}{8} - \frac{9}{4}{\sin ^{ - 1}}\frac{1}{3} = \frac{9}{4}{\sin ^{ - 1}}\frac{{2\sqrt 2 }}{3}$ SA Q13 $2{\tan ^{ - 1}}(\cos x) = {\tan ^{ - 1}}(2{\rm{cosec }}x)$ SA Q14 ${\tan ^{ - 1}}\left( {\frac{{1 - x}}{{1 + x}}} \right) = \frac{1}{2}{\tan ^{ - 1}}x,\;\;(x > 0)$ SA Q15 $\sin ({\tan ^{ - 1}}x),\;|x|\; < 1$ is equal to

(A) $\frac{x}{{\sqrt {1 - {x^2}} }}$

(B) $\frac{1}{{\sqrt {1 - {x^2}} }}$

(C) $\frac{1}{{\sqrt {1 + {x^2}} }}$

(D) $\frac{x}{{\sqrt {1 + {x^2}} }}$ SA Q16 . If ${\sin ^{ - 1}}(1 - x) - 2{\sin ^{ - 1}}x = \frac{\pi }{2},\;\;then\;\;x$ is equal to

(A) $0,\;\;\frac{1}{2}$

(B) $1,\;\;\frac{1}{2}$

(C) 0

(D) $\frac{1}{2}$ SA Q17 ${\tan ^{ - 1}}\left( {\frac{x}{y}} \right) - {\tan ^{ - 1}}\left( {\frac{{x - y}}{{x + y}}} \right)$ is equal to

(A) $\frac{\pi }{2}$

(B) $\frac{\pi }{3}$

(C) $\frac{\pi }{4}$

(D) $\frac{{ - 3\pi }}{4}$ SA