Question

Solve the following equation:
3tanx + cot x = 5 cosec x

Answer 1

\[3 \tan x + \cot x = 5 cosec x\]
\[ \Rightarrow \frac{3 \sin x}{\cos x} + \frac{\cos x}{\sin x} = \frac{5}{\sin x}\]
\[ \Rightarrow \frac{3 \sin^2 x + \cos^2 x}{\cos x \sin x} = \frac{5}{\sin x}\]
\[ \Rightarrow 3\left( 1 - \cos^2 x \right) + \cos^2 x = 5 \cos x\]
\[ \Rightarrow 3 - 3 \cos^2 x + \cos^2 x = 5 \cos x\]
\[ \Rightarrow 2 \cos^2 x + 5 \cos x - 3 = 0\]
\[ \Rightarrow 2 \cos^2 x + 6 \cos x - \cos x - 3 = 0\]
\[ \Rightarrow 2 \cos x\left( \cos x + 3 \right) - 1\left( \cos x + 3 \right) = 0\]
\[ \Rightarrow \left( 2 \cos x - 1 \right)\left( \cos x + 3 \right) = 0\]
\[ \Rightarrow \left( 2 \cos x - 1 \right) = 0\text{ or }\left( \cos x + 3 \right) = 0\]
\[ \Rightarrow \cos x = \frac{1}{2}\text{ or }\cos x = - 3\]
\[\cos x = - 3\text{ is not possible }\left( \because - 1 \leq \cos x \leq 1 \right)\]
\[ \Rightarrow \cos x = \cos\frac{\pi}{3}\]
\[ \Rightarrow x = 2n\pi \pm \frac{\pi}{3}, n \in \mathbb{Z}\]