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Question
Solve the following equation:
sin x tan x – 1 = tan x – sin x
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Solution
\[\sin x \tan x - 1 = \tan x - \sin x\]
\[ \Rightarrow \sin x \tan x - \tan x + \sin x - 1 = 0\]
\[ \Rightarrow \tan x\left( \sin x - 1 \right) + 1\left( \sin x - 1 \right) = 0\]
\[ \Rightarrow \left( \tan x + 1 \right)\left( \sin x - 1 \right) = 0\]
\[ \Rightarrow \left( \tan x + 1 \right) = 0\text{ or }\left( \sin x - 1 \right) = 0\]
\[ \Rightarrow \tan x = - 1\text{ or }\sin x = 1\]
\[ \Rightarrow \tan x = \tan\frac{3\pi}{4}\text{ or }\sin x = \sin\frac{\pi}{2}\]
\[ \Rightarrow x = n\pi + \frac{3\pi}{4}\text{ or }x = n\pi + \left( - 1 \right)^n \frac{\pi}{2}, n \in \mathbb{Z}\]
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