Prove that \(\tan^{-1}\left(\dfrac{\sqrt{1+x^{2}} + \sqrt{1-x^{2}}}{\sqrt{1+x^{2}} - \sqrt{1-x^{2}}}\right) = \dfrac{\pi}{4} + \dfrac{1}{2} \cos^{-1} x^{2}\).
Find the simplified form of \(\cos^{-1}\left(\dfrac{3}{5}\cos x + \dfrac{4}{5}\sin x\right)\), where \(x \in \left[ -\dfrac{3\pi}{4}, \dfrac{\pi}{4} \right]\).
\(\tan^{-1}\dfrac{4}{3} - x\)
Prove that \(\sin^{-1}\dfrac{8}{17} + \sin^{-1}\dfrac{3}{5} = \sin^{-1}\dfrac{77}{85}\).
\(\dfrac{77}{85}\)
Show that \(\sin^{-1}\dfrac{5}{13} + \cos^{-1}\dfrac{3}{5} = \tan^{-1}\dfrac{63}{16}\).
\(\dfrac{63}{16}\)
Prove that \(\tan^{-1}\dfrac{1}{4} + \tan^{-1}\dfrac{2}{9} = \sin^{-1}\dfrac{1}{\sqrt{5}}\).
\(\dfrac{1}{\sqrt{5}}\)
Find the value of \(4 \tan^{-1}\dfrac{1}{5} - \tan^{-1}\dfrac{1}{239}\).
\(\dfrac{\pi}{4}\)
Show that \(\tan\left(\dfrac{1}{2}\sin^{-1}\dfrac{3}{4}\right) = \dfrac{4 - \sqrt{7}}{3}\) and justify why the other value \(\dfrac{4 + \sqrt{7}}{3}\) is ignored.
\(\dfrac{4 - \sqrt{7}}{3}\)
If \(a_{1}, a_{2}, a_{3}, \ldots, a_{n}\) is an arithmetic progression with common difference \(d\), evaluate the expression:
\[ \tan \left[ \tan^{-1}\left(\dfrac{d}{1 + a_{1}a_{2}}\right) + \tan^{-1}\left(\dfrac{d}{1 + a_{2}a_{3}}\right) + \tan^{-1}\left(\dfrac{d}{1 + a_{3}a_{4}}\right) + \cdots + \tan^{-1}\left(\dfrac{d}{1 + a_{n-1}a_{n}}\right) \right] \]
\(\dfrac{a_{n} - a_{1}}{1 + a_{1}a_{n}}\)