With practice, solving trigonometric equations becomes systematic. Memorize the general solution forms and always check your solutions in the original equation.
Case 1: ( \sin x = 0 \Rightarrow x = 0, \pi ) in ( [0, 2\pi) ). Case 2: ( \cos x = 1/2 \Rightarrow x = \pi/3,\ 5\pi/3 ) in ( [0, 2\pi) ).
Method 1: Divide by ( \cos x ) (if ( \cos x \neq 0 )): ( \tan x = 1 \Rightarrow x = \pi/4 + k\pi ). Check ( \cos x = 0 ) gives ( x = \pi/2, 3\pi/2 ), but those don’t satisfy original (sine ≠ cosine). So fine.
( 2\sin x = 1 ) ( \cos^2 x - \sin^2 x = 0 ) ( \tan(2x) = \sqrt3 )
