Solve tan2x=cot2x in the range 0-180

Tan (2x) - 1/Tan (2x) = 0 / / clear the denominators by multiplying both sides by Tan (2x)

Tan² (2x) - 1 = 0 as long as Tan (2x) exists and ≠ 0

Tan² (2x) = 1 so Tan (2x) = - 1 or Tan (2x) = 1

Therefore, 2x = 45° or x = 22.5° or 2x = - 45° or x = - 22.5°

There are many other values of x that work but not between - 180° and 180°.