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

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

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tan^2(θ)+3sec(θ)=-3

tan^{2} (θ) + 3 sec (θ) = - 3

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

Solution steps

tan^{2} (θ) + 3 sec (θ) = - 3

Subtract

- 3

from both sides

tan^{2} (θ) + 3 sec (θ) + 3 = 0

Rewrite using trig identities

2 + sec^{2} (θ) + 3 sec (θ) = 0

Solve by substitution

sec (θ) = - 1, sec (θ) = - 2

sec (θ) = - 1 : θ = π + 2 πn

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

Combine all the solutions

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

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

sin (2 x) cos (x) + sin (x) = 0

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

- sin (θ) + 2 sin^{2} (θ) = 0

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

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