Question

find all SOLUTIONS of the following equation in the interval [0,2π)

tan x- cot x=0

Answer 1

tan x = sin x/cos x
cot x = 1/tanx

therefore the equation reduces to:
\[\frac{sin x}{cos x}=\frac{cos x}{sin x}\]
and thus
\[cos^2 x - sin^2 x = 0\]
using the identity:
\[cos^2 x - sin^2 x = cos2x\]
It becomes:
\[cos2x=0\]

Thus,
\[2x=\frac{\pi}{2}+n\pi\]