What is the general solution for tan^2(x)+2sec(x)=-1 ?

The general solution for tan^2(x)+2sec(x)=-1 is x=(2pi)/3+2pin,x=(4pi)/3+2pin

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tan^2(x)+2sec(x)=-1

tan^{2} (x) + 2 sec (x) = - 1

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

Solution steps

tan^{2} (x) + 2 sec (x) = - 1

Subtract

- 1

from both sides

tan^{2} (x) + 2 sec (x) + 1 = 0

Rewrite using trig identities

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

Solve by substitution

sec (x) = 0, sec (x) = - 2

sec (x) = 0 :

No Solution

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

Combine all the solutions

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

2 cos^{2} (θ) + 3 cos (θ) + 1 = 0

tan^{2} (x) sin (x) = - sin (x)

cos (4 x) = sin (x), 0 \leq x \leq 2 π

2 cos (x) = 1

sin (θ) = \frac{- 3}{2}

What is the general solution for tan^2(x)+2sec(x)=-1 ?
The general solution for tan^2(x)+2sec(x)=-1 is x=(2pi)/3+2pin,x=(4pi)/3+2pin