Prove that $\tan^{-1}\left(\frac{1}{2}\right) + \tan^{-1}\left(\frac{1}{3}\right) = \frac{\pi}{4}$.

Prove that $4\tan^{-1}\left(\frac{1}{5}\right) - \tan^{-1}\left(\frac{1}{239}\right) = \frac{\pi}{4}$ (Machin's Formula).

Find the general solution of the trigonometric equation $\sqrt{3}\cos\theta + \sin\theta = \sqrt{2}$.

Evaluate the principal value of each of the following inverse circular functions: (a) $\arcsin\left(\frac{1}{2}\right)$ (b) $\arccos\left(-\frac{\sqrt{3}}{2}\right)$ (c) $\arctan(-1)$ (d) $\text{arcsec}(-2)$