Question

Solve the equation cot x + tan x = 2 cosecx

Answer 1

As
$tan \: x = \frac{1}{cot \: x}$
put this value in the equation

$cot \: x + \frac{1}{cot \: x} = 2 \: cosec \: x \\ \\ \frac{ {cot}^{2}x \: + 1 }{cot \: x} = 2cosec \: x$

As

$1 + {cot}^{2} x = {cosec}^{2} x$

$\frac{ {cosec}^{2} x}{cot \: x} = 2 \: cosec \: x \\ \\ cosec \: x = 2cot \: x \\ \\ \frac{1}{sin \: x} = 2 (\frac{cos \: x}{sin \: x} ) \\ \\ 2 \: cos \: x = 1 \\ \\ cos \: x = \frac{1}{2} \\ \\ x = {cos}^{ - 1} ( \frac{1}{2} ) \\ \\ x = {cos}^{ - 1} (cos \: ( \frac{ \pi}{3} ) \\ \\ x = \frac{\pi}{3}$

Since this belongs to principal value of cos -1 x.

Answer 2

Another method to do the same :)