What is the general solution for tan^2(x)+3sec(x)=-3 ?

The general solution for tan^2(x)+3sec(x)=-3 is x=pi+2pin,x=(2pi)/3+2pin,x=(4pi)/3+2pin

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

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

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

Solution steps

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

Subtract

- 3

from both sides

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

Rewrite using trig identities

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

Solve by substitution

sec (x) = - 1, sec (x) = - 2

sec (x) = - 1 : x = π + 2 πn

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

Combine all the solutions

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

csc (x) = - \frac{2 3}{3}

sin (2 x) = - 5 cos (2 x), 0 \leq x \leq 2 π

2 cos^{2} (x) + 9 cos (x) - 5 = 0

cos (θ) = - \frac{3}{4}

tan (θ) = \frac{3}{2}

What is the general solution for tan^2(x)+3sec(x)=-3 ?
The general solution for tan^2(x)+3sec(x)=-3 is x=pi+2pin,x=(2pi)/3+2pin,x=(4pi)/3+2pin