Trig Question Thank You In Advance

Question

Solve for x by factoring where pi/2 ≤ x ≤ pi.

3 csc x cot^2 x = csc x

Answer 1

3 csc x cot^2 x = csc x

3 csc x cot^2 x - csc x = 0

csc(x)[3cot²(x) - 1] = 0

This only can happen if the two factors equal 0 so:

csc(x) = 0 (no solution because csc(x) is always > 1)

and

3cot²(x) - 1 = 0

3cot²(x) = 1

cot²(x) = 1/3

cot(x) = ± 1/√3

Looking at the unit circle, the only solution between π/2 and π is when cot(x) = -1/√3 at 2π/3

cot(2π/3) = adj/opp = (-1/2)/(√3/2) = -1/√3