What is the general solution for tan^2(θ)+4sec(θ)=-5 ?

The general solution for tan^2(θ)+4sec(θ)=-5 is θ=(2pi)/3+2pin,θ=(4pi)/3+2pin

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tan^2(θ)+4sec(θ)=-5

tan^{2} (θ) + 4 sec (θ) = - 5

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

Solution steps

tan^{2} (θ) + 4 sec (θ) = - 5

Subtract

- 5

from both sides

tan^{2} (θ) + 4 sec (θ) + 5 = 0

Rewrite using trig identities

4 + sec^{2} (θ) + 4 sec (θ) = 0

Solve by substitution

sec (θ) = - 2

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

Combine all the solutions

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

0 = tan (θ)

cos (θ) = \frac{4}{9}

- 6 cos (x) = 0

tan (x) 3 cot (x) = 5 sec (x)

10 cos (θ) = 5

What is the general solution for tan^2(θ)+4sec(θ)=-5 ?
The general solution for tan^2(θ)+4sec(θ)=-5 is θ=(2pi)/3+2pin,θ=(4pi)/3+2pin