What is the general solution for tan^2(x)+3sec(x)+3=0 ?

The general solution for tan^2(x)+3sec(x)+3=0 is x=pi+2pin,x=(2pi)/3+2pin,x=(4pi)/3+2pin

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tan^2(x)+3sec(x)+3=0

tan^{2} (x) + 3 sec (x) + 3 = 0

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

Solution steps

tan^{2} (x) + 3 sec (x) + 3 = 0

Rewrite using trig identities

2 + sec^{2} (x) + 3 sec (x) = 0

Solve by substitution

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

sec (x) = - 1 : x = π + 2 πn

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

Combine all the solutions

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

4 cos (x) - 3 sin (x) = 0

cos (\frac{π}{4} - x) = 1

sin (b) = \frac{1}{2}

2 cos (θ) = 0

3 tan (θ) = 2 cos (θ)

What is the general solution for tan^2(x)+3sec(x)+3=0 ?
The general solution for tan^2(x)+3sec(x)+3=0 is x=pi+2pin,x=(2pi)/3+2pin,x=(4pi)/3+2pin