Analytic Trigonometry

Question

Please explain steps.

Find all solutions to the equation.

tan3x+1=sec3x

Answer 1

tan(3x) + 1 = sec(3x)

Apply square each side.

[tan(3x) + 1]² = [sec(3x)]²

tan²(3x)+ 1 + 2tan(3x) = sec²(3x)

Pythagorean Identities: tan²θ+1 = sec²θ

sec²(3x) + 2tan(3x) = sec²(3x)

Subtract sec²(3x) from each side.

2tan(3x) = 0

Divide each side by 2.

tan(3x) = 0

3x = arc tan(0) = 0, 2π, 4π..............

Divide each side by 3.

x = 0, 2π/3, 4π/3, ..............

Therefore x = 0, 2π/3, 4π/3, ..............