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

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

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

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

x = \frac{3 π}{4} + πn

Solution steps

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

Subtract

2 tan (x)

from both sides

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

Rewrite using trig identities

- 1 - tan^{2} (x) - 2 tan (x) = 0

Solve by substitution

tan (x) = - 1

tan (x) = - 1 : x = \frac{3 π}{4} + πn

Combine all the solutions

x = \frac{3 π}{4} + πn

3 cot^{2} (x) - 1 = 0

sin (x) \cdot cos (x) = 1

(2 cos^{2} (x) - 1) (4 sin^{2} (x) - 3) = 0

sin^{2} (t) = 1

4 cos (x) sin (x) = 1

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