List of top Mathematics Questions on limits of trigonometric functions

Find the value of the following expression: \[ \tan^2(\sec^{-1}4) + \cot(\csc^{-1}3) \]

Given a real-valued function \( f \) such that: \[ f(x) = \begin{cases} \frac{\tan^2\{x\}}{x^2 - \lfloor x \rfloor^2}, & \text{for } x > 0 \\ 1, & \text{for } x = 0 \\ \sqrt{\{x\} \cot\{x\}}, & \text{for } x < 0 \end{cases} \] Then:

Evaluate
\[ \lim_{x \to 0} \sqrt{\frac{x - \sin x}{x + \sin^2 x}} \]

\[ \lim_{x \to \infty} \frac{1 - \tan\left(\frac{x}{2}\right)}{1 + \tan\left(\frac{x}{2}\right)} = ? \]

\( \limₓ → ∞ \frac1 - \tan \left( \fracx2 \right)1 + \tan \left( \fracx2 \right) = ?

If sin $5x+\sin 3x+\sin x = 0$, then the value of x other than zero, lying between π$0\le \times \le \frac{\mathrm{\pi }}{2}$ is;