3sec^2(x)-2tan^2(x)=4

Solution

3 sec^{2} (x) - 2 tan^{2} (x) = 4

x = \frac{π}{4} + 2 πn, x = \frac{7 π}{4} + 2 πn, x = \frac{3 π}{4} + 2 πn, x = \frac{5 π}{4} + 2 πn

Solution steps

3 sec^{2} (x) - 2 tan^{2} (x) = 4

Subtract

4

from both sides

3 sec^{2} (x) - 2 tan^{2} (x) - 4 = 0

Rewrite using trig identities

- 2 + sec^{2} (x) = 0

Solve by substitution

sec (x) = 2, sec (x) = - 2

sec (x) = 2 : x = \frac{π}{4} + 2 πn, x = \frac{7 π}{4} + 2 πn

sec (x) = - 2 : x = \frac{3 π}{4} + 2 πn, x = \frac{5 π}{4} + 2 πn

Combine all the solutions

x = \frac{π}{4} + 2 πn, x = \frac{7 π}{4} + 2 πn, x = \frac{3 π}{4} + 2 πn, x = \frac{5 π}{4} + 2 πn

sin (x) = \frac{1}{4}

2 sin^{2} (x) - 5 sin (x) + 2 = 0

sin (x + \frac{π}{3}) + sin (x - \frac{π}{3}) = 1

sec^{2} (x) = 4

1 + 2 cos (x) = 0