Question

2 cos a cot^2 a - 6 cos a - cot^2 a = -3 is it an identity or a conditional equation? if not an identity, solve for the unknown

Answer 1

The given equation $2cosa.cot^2a-6cosa-cot^2a=-3$ is identity equation.

Solving the above equation,

$2cosa.cot^2a-6cosa-cot^2a=-3\newline 2cot^2a.cosa-cot^2a+3-6cosa=0\newline cot^2a(2cosa-1)-3(2cosa-1)=0\newline (cot^2a-3)(2cosa-1)=0\newline (cota+ \sqrt3)(cota-\sqrt3)(2cosa-1)=0$

Then, we get

$cota+ \sqrt3=0..............................1\newline cota-\sqrt3=0..............................2\newline 2cosa-1=0................................3$

General solution of 1 is $a=n\pi +\frac{5 \pi }{6}$ , 2 is $a=n\pi +\frac{\pi }{6}$ and 3 is $a=2n\pi +\frac{\pi }{3} \text{and}\space a=2n\pi +\frac{5\pi }{3}$.

Thus, the required solution are $a=n\pi +\frac{5 \pi }{6}$, $a=n\pi +\frac{\pi }{6}$, $a=2n\pi +\frac{\pi }{3} \text{and}\space a=2n\pi +\frac{5\pi }{3}$.