List of top Mathematics Questions on Trigonometry asked in TS EAMCET

If \[ \int \sqrt{\csc x + 1} \, dx = k \tan^{-1} (f(x)) + c, \] then \[ \frac{1}{k} f\left(\frac{\pi}{6}\right) = ? \]
The trigonometric equation \( \sin^{-1}x = 2\sin^{-1}a \) has a solution. Find the valid range for \( a \).