a.math

solve the following equation for 0≦χ≦360
tanχ - cotχ=1-cot2χ
tanχ - cotχ = 1 - cot2χ
tanχ - (1/tanχ) = 1 - (1/tan2χ)
tanχ - (1/tanχ) = 1 - [1/[2tanχ/(1-tan2χ)]]
【Using identity tan2A = (2tanA)/(1-tan2A)】
tanχ - (1/tanχ) = 1 - (1-tan2χ) / 2tanχ
2tan2χ - 2 = 2tanχ - (1 - tan2χ) 【Multiply 2tanχ in both sides】
2tan2χ - 2 = 2tanχ - 1 + tan2χ
tan2χ - 2tanχ - 1 = 0
tan2χ - 2tanχ + 1 = 2
(tanχ - 1)2 = 2
tanχ - 1 = √2 or tanχ - 1 = -√2
tanχ = √2 + 1 or tanχ = -√2 + 1
χ = 67.5° or χ = 247.5° or χ = 157.5° or χ = 337.5°