Q.Find the principal solutions of the equation $\tan 5\theta = -1$

A trigonometric equation is any equation involving one or more trigonometric functions of an unknown angle, such as $\sin\theta=\frac12$ or $\cos3\theta=\cos5\theta$. Because every trigonometric function is periodic, such an equation is satisfied by infinitely many angles rather than a finite list of numbers — once one solution is found, adding any whole-number multiple of the function's period gives another. To make 'the solutions' of an equation a manageable, well-defined set, we agree to first report only the solutions lying in one fixed window, the interval $[0,2\pi)$; these are called the principal solutions. Finding them typically means recognising the equation's right-hand side as the value of a standard angle (like $\frac{\pi}{6}$, $\frac{\pi}{4}$, $\frac{\pi}{3}$), and then using an allied-angle identit …