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