Question

Find the principal solution of the equation.
tan x = -1

Answer 1

We know that principal value branch of

${tan}^{ - 1} x \:\: is\:\: [-\frac{\pi}{2}, \frac{\pi}{2}]$

To solve the given equation we must keep consider that tan and tan inverse cancels each other only if x belongs to its principal value .

$tan \: x = -1\\ \\ x = {tan}^{ - 1} (-1) \\ \\ we \: know \: that \: tan \: \frac{\pi}{4} = 1\\ \\We\:\: know\:\:that\:\:\\\\tan \: ( - x) = - tan \: x \\ \\ {tan}^{ - 1} ( - tan \: x) = \pi -{tan}^{ - 1} ( tan \: x) \\\\ So\\\\ x= {tan}^{ - 1} (tan ( -\frac{\pi}{4} )) \\ \\x=\pi- {tan}^{ - 1} (tan ( \frac{\pi}{4} ))\\\\ x= \pi - \frac{ \pi}{4} \: \: \: \: (x \: belongs \: to \: [- \frac{\pi}{2}, \frac{\pi}{2}]) \\ \\ x = \frac{3\pi}{4}$

is the solution of the equation.