solvefor x,2sec^2(x)+tan^2(x)-3=0

Solution

solve for

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

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

Solution steps

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

Rewrite using trig identities

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

Solve by substitution

sec (x) = \frac{2 3}{3}, sec (x) = - \frac{2 3}{3}

sec (x) = \frac{2 3}{3} : x = \frac{π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

sec (x) = - \frac{2 3}{3} : x = \frac{5 π}{6} + 2 πn, x = \frac{7 π}{6} + 2 πn

Combine all the solutions

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

5 sin (\frac{θ}{2}) = 5 cos (\frac{θ}{2})

tan (x) = \frac{80}{50}

- cos (x) + 4 cos (2 x) = 0

sin^{2} (x) + sin (x) - 2 = 0, 0 \leq x < 2 π

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