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

Solution

2 csc^{2} (x) = 3 cot^{2} (x) - 1

x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn, x = \frac{7 π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

Solution steps

2 csc^{2} (x) = 3 cot^{2} (x) - 1

Subtract

3 cot^{2} (x) - 1

from both sides

2 csc^{2} (x) - 3 cot^{2} (x) + 1 = 0

Rewrite using trig identities

4 - csc^{2} (x) = 0

Solve by substitution

csc (x) = 2, csc (x) = - 2

csc (x) = 2 : x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

csc (x) = - 2 : x = \frac{7 π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

Combine all the solutions

x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn, x = \frac{7 π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

cot^{2} (x) = 3

3 cos (x) = sin (x)

- 2 sin^{2} (x) + cos (x) + 1 = 0

sin (x) tan (x) = 2 tan (x)

tan^{2} (x) - 2 tan (x) = 0