Find the principal value of \( \sin^{-1}\left(-\dfrac{1}{2}\right) \).
\( -\dfrac{\pi}{6} \)
Find the principal value of \( \cos^{-1}\left(\dfrac{\sqrt{3}}{2}\right) \).
\( \dfrac{\pi}{6} \)
Find the principal value of \( \text{cosec}^{-1}(2) \).
\( \dfrac{\pi}{6} \)
Find the principal value of \( \tan^{-1}(-\sqrt{3}) \).
\( -\dfrac{\pi}{3} \)
Find the principal value of \( \cos^{-1}\left(-\dfrac{1}{2}\right) \).
\( \dfrac{2\pi}{3} \)
Find the principal value of \( \tan^{-1}(-1) \).
\( -\dfrac{\pi}{4} \)
Find the principal value of \( \sec^{-1}\left(\dfrac{2}{\sqrt{3}}\right) \).
\( \dfrac{\pi}{6} \)
Find the principal value of \( \cot^{-1}(\sqrt{3}) \).
\( \dfrac{\pi}{6} \)
Find the principal value of \( \cos^{-1}\left(-\dfrac{1}{\sqrt{2}}\right) \).
\( \dfrac{3\pi}{4} \)
Find the principal value of \( \text{cosec}^{-1}(-\sqrt{2}) \).
\( -\dfrac{\pi}{4} \)
Find the value of \( \tan^{-1}(1) + \cos^{-1}\left(-\dfrac{1}{2}\right) + \sin^{-1}\left(-\dfrac{1}{2}\right) \).
\( \dfrac{3\pi}{4} \)
Find the value of \( \cos^{-1}\left(\dfrac{1}{2}\right) + 2\sin^{-1}\left(\dfrac{1}{2}\right) \).
\( \dfrac{2\pi}{3} \)
If \( \sin^{-1} x = y \), then which of the following is correct?
(A) \( 0 \le y \le \pi \)
(B) \( -\dfrac{\pi}{2} \le y \le \dfrac{\pi}{2} \)
(C) \( 0 < y < \pi \)
(D) \( -\dfrac{\pi}{2} < y < \dfrac{\pi}{2} \)
B
Find the value of \( \tan^{-1}(\sqrt{3}) - \sec^{-1}(-2) \).
(A) \( \pi \)
(B) \( -\dfrac{\pi}{3} \)
(C) \( \dfrac{\pi}{3} \)
(D) \( \dfrac{2\pi}{3} \)
B
Prove the following:
\( 3\sin^{-1}x = \sin^{-1}(3x - 4x^3), \; x \in \left[-\dfrac{1}{2}, \dfrac{1}{2}\right] \).
Identity holds as given.
Prove the following:
\( 3\cos^{-1}x = \cos^{-1}(4x^3 - 3x), \; x \in \left[\dfrac{1}{2}, 1\right] \).
Identity holds as given.
Write the following function in simplest form:
\( \tan^{-1}\left(\dfrac{\sqrt{1+x^2}-1}{x}\right), \; x \neq 0 \).
\( \dfrac{1}{2}\tan^{-1}x \)
Write the following function in simplest form:
\( \tan^{-1}\left(\sqrt{\dfrac{1 - \cos x}{1 + \cos x}}\right), \; 0 < x < \pi \).
\( \dfrac{x}{2} \)
Write the following function in simplest form:
\( \tan^{-1}\left(\dfrac{\cos x - \sin x}{\cos x + \sin x}\right), -\dfrac{\pi}{4} < x < \dfrac{3\pi}{4} \).
\( \dfrac{\pi}{4} - x \)
Write the following function in simplest form:
\( \tan^{-1}\left(\dfrac{x}{\sqrt{a^2 - x^2}}\right), \; |x| < a \).
\( \sin^{-1}\left(\dfrac{x}{a}\right) \)
Write the following function in simplest form:
\( \tan^{-1}\left(\dfrac{3a^2x - x^3}{a^3 - 3ax^2}\right), a > 0, \; -\dfrac{a}{\sqrt{3}} < x < \dfrac{a}{\sqrt{3}} \).
\( 3\tan^{-1}\left(\dfrac{x}{a}\right) \)
Find the value of:
\( \tan\left[2\cos\left(2\sin^{-1}\dfrac{1}{2}\right)\right] \).
\( \dfrac{\pi}{4} \)
Find the value of:
\( \tan\left[\dfrac{1}{2}\left(\sin^{-1}\dfrac{2x}{1+x^2} + \cos^{-1}\dfrac{1-y^2}{1+y^2}\right)\right], |x|<1, y>0, xy<1 \).
\( \dfrac{x + y}{1 - xy} \)
Find the value of:
\( \sin^{-1}\left(\sin\dfrac{2\pi}{3}\right) \).
\( \dfrac{\pi}{3} \)
Find the value of:
\( \tan^{-1}\left(\tan\dfrac{3\pi}{4}\right) \).
\( -\dfrac{\pi}{4} \)
Find the value of:
\( \tan\left(\sin^{-1}\dfrac{3}{5} + \cot^{-1}\dfrac{3}{2}\right) \).
\( \dfrac{17}{6} \)
Find the value of:
\( \cos^{-1}(\cos\dfrac{7\pi}{6}) \).
(A) \( \dfrac{7\pi}{6} \)
(B) \( \dfrac{5\pi}{6} \)
(C) \( \dfrac{\pi}{3} \)
(D) \( \dfrac{\pi}{6} \)
B
Find the value of:
\( \sin\left(\dfrac{\pi}{3} - \sin^{-1}\left(-\dfrac{1}{2}\right)\right) \).
(A) \( \dfrac{1}{2} \)
(B) \( \dfrac{1}{3} \)
(C) \( \dfrac{1}{4} \)
(D) 1
D
Find the value of:
\( \tan^{-1}(\sqrt{3}) - \cot^{-1}(-\sqrt{3}) \).
(A) \( \pi \)
(B) \( -\dfrac{\pi}{2} \)
(C) 0
(D) \( 2\sqrt{3} \)
B
Find the value of the following:
\( \cos^{-1}(\cos \frac{13\pi}{6}) \)
\( \frac{\pi}{6} \)
Find the value of the following:
\( \tan^{-1}(\tan \frac{7\pi}{6}) \)
\( \frac{\pi}{6} \)
Prove that:
\( 2\sin^{-1}\frac{3}{5} = \tan^{-1}\frac{24}{7} \)
Identity holds as given.
Prove that:
\( \sin^{-1}\frac{8}{17} + \sin^{-1}\frac{3}{5} = \tan^{-1}\frac{77}{36} \)
Identity holds as given.
Prove that:
\( \cos^{-1}\frac{4}{5} + \cos^{-1}\frac{12}{13} = \cos^{-1}\frac{33}{65} \)
Identity holds as given.
Prove that:
\( \cos^{-1}\frac{12}{13} + \sin^{-1}\frac{3}{5} = \sin^{-1}\frac{56}{65} \)
Identity holds as given.
Prove that:
\( \tan^{-1}\frac{63}{16} = \sin^{-1}\frac{5}{13} + \cos^{-1}\frac{3}{5} \)
Identity holds as given.
Prove that:
\( \tan^{-1}\sqrt{x} = \frac{1}{2}\cos^{-1}\frac{1-x}{1+x}, \; x \in [0,1] \)
Identity holds as given.
Prove that:
\( \cot^{-1}\left(\sqrt{\frac{1+\sin x}{1-\sin x}}\right) = \frac{x}{2}, \; x \in \left(0,\frac{\pi}{4}\right) \)
Identity holds as given.
Prove that:
\( \tan^{-1}\left(\sqrt{\frac{1+x}{1-x}}\right) = \frac{\pi}{4} + \frac{1}{2}\cos^{-1}x, \; -\frac{1}{\sqrt{2}} \le x \le 1 \)
Identity holds as given.
Solve the equation:
\( 2\tan^{-1}(\cos x) = \tan^{-1}(2\csc x) \)
\( x = n\pi + \frac{\pi}{4}, \; n \in \mathbb{Z} \)
Solve the equation:
\( \tan^{-1}\frac{1-x}{1+x} = \frac{1}{2}\tan^{-1}x, \; x > 0 \)
\( x = \frac{1}{\sqrt{3}} \)
\( \sin(\tan^{-1}x), |x| < 1 \) 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) \( \frac{x}{\sqrt{1+x^2}} \)
D
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) \( \frac{1}{2} \)
C