Question

Find the general solution of the following equation:

\[\sin x = \tan x\]

Answer 1

We have:

\[\sin x = \tan x\]

\[\Rightarrow \sin x - \tan x = 0\]
\[ \Rightarrow \sin x - \frac{\sin x}{\cos x} = 0\]
\[ \Rightarrow \sin x \left( 1 - \frac{1}{\cos x} \right) = 0\]
\[ \Rightarrow \sin x (\cos x - 1) = 0\]

\[\Rightarrow \sin x = 0\] or

\[\cos x - 1 = 0\]

Now,  

\[\sin x = 0 \Rightarrow x = n\pi, n \in Z\]

\[\cos x - 1 = 0 \]

\[ \Rightarrow \cos x = 1 \]

\[ \Rightarrow \cos x = \cos0 \]

\[ \Rightarrow x = 2m\pi, m \in Z\]