solve for all 0≤x≤2π

Question

1. 2tan²x-3sec x+3=0
2. cot x-√3=csc x

Answer 1

We can use a pythagorean trig identity to change equation 1. into a quadratic in secx, because tan²x = sec²x - 1:

2sec²x - 3secx + 1 = 0

(2secx - 1)(secx - 1) = 0

secx = 1/2 (no solutions) or secx = 1 ; x = 0 (or 2π , but often the restriction is 0 ≤ x < 2π)

cotx - cscx = √3

(cosx - 1)/sinx = √3

cosx - 1 = √3·sinx

cos²x - 2cosx + 1 = 3sin²x

cos²x - 2cosx + 1 = 3(1 - cos²x)

4cos²x - 2cosx - 2 = 0

2cos²x - cosx - 1 = 0

(2cosx + 1)(cosx - 1) = 0

cosx = -1/2 (x = 2π/3 or 4π/3) or cosx = 1 (no solns, since cot0 and csc0 undefined)