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

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

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

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

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

Solution steps

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

Subtract

1 - sec (x)

from both sides

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

Rewrite using trig identities

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

Solve by substitution

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

sec (x) = 1 : x = 2 πn

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

Combine all the solutions

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

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

cos (4 x) - cos (2 x) = 0

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

2 cos (θ) = - 2

tan (\frac{x}{3}) = 3

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