sec^2(x)-2tan^2(x)=0

Solution

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

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

Solution steps

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

Factor

sec^{2} (x) - 2 tan^{2} (x) : (sec (x) + 2 tan (x)) (sec (x) - 2 tan (x))

(sec (x) + 2 tan (x)) (sec (x) - 2 tan (x)) = 0

Solving each part separately

sec (x) + 2 tan (x) = 0 or sec (x) - 2 tan (x) = 0

sec (x) + 2 tan (x) = 0 : x = \frac{5 π}{4} + 2 πn, x = \frac{7 π}{4} + 2 πn

sec (x) - 2 tan (x) = 0 : x = \frac{π}{4} + 2 πn, x = \frac{3 π}{4} + 2 πn

Combine all the solutions

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

tan (5 x) + 3 = 0

2 tan (x) sin (x) - 7 sin (x) = 0

sin^{2} (x) + sin (x) = 2

- 3 sin^{2} (θ) + 3 = - 10 cos (θ) - 3

cos (2 x) + 2 sin (x) = 1