Question

how do i solve this using trig identities?

2 csc^2x = 3 cot^2x-1

Answer 1

first change everything to sines and cosines
2/(sin^2 x) = 3(cos^2 x)/(sin^2 x) - 1
multiply by sin^2 x
2 = 3 cos^2 x - sin^2 x
but sin^2 x - 1 - cos^2 x
so
2 = 3 cos^2 x - 1 + cos^2 x
3 = 4cos^2 x
cos^2 x = 3/4
cos x = ±√3/2
Do you recognize the 1,√3,2 right-angled triangle? If not your calculuator should give you 30º.

if cos x = +√3/2 then x is in the first our fourth quadrants and
x = 30º or 330º

if cos x = -√3/2 then x is in the second and third quadrants and
x = 150º or 210º

(in radians the answers would be
pi/6, 11pi/6, 5pi/6,7pi/6)