Considering only the principal values of the inverse trigonometric functions, the value of \( \tan\!\left(\sin^{-1}\!\left(\frac{3}{5}\right)-2\cos^{-1}\!\left(\frac{2}{\sqrt{5}}\right)\right) \) is
\( \frac{7}{24} \)
\( \frac{-7}{24} \)
\( \frac{-5}{24} \)
\( \frac{5}{24} \)
For any positive integer \( n \), let \( S_{n} : (0,\infty) \to \mathbb{R} \) be defined by \( S_{n}(x) = \displaystyle \sum_{k=1}^{n} \cot^{-1}\!\left(\dfrac{1 + k(k+1)x^{2}}{x}\right) \)Where for any \( x \in \mathbb{R} \), \( \cot^{-1}(x) \in (0,\pi) \) and \( \tan^{-1}(x) \in \left( -\dfrac{\pi}{2}, \dfrac{\pi}{2} \right) \). Then which of the following statements is (are) TRUE?
Considering only the principal values of the inverse trigonometric functions, the value of\( \cot^{-1}(\cot(-11)) + 10\sin\!\left(2\cos^{-1}\!\left(\frac{1}{\sqrt{2}}\right)\right) + 10\sin\!\left(2\tan^{-1}(2)\right) \) is:
