What is the general solution for 4cos^2(x)+3sin^2(x)-4=-tan^2(x) ?

The general solution for 4cos^2(x)+3sin^2(x)-4=-tan^2(x) is x=2pin,x=pi+2pin

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4cos^2(x)+3sin^2(x)-4=-tan^2(x)

4 cos^{2} (x) + 3 sin^{2} (x) - 4 = - tan^{2} (x)

x = 2 πn, x = π + 2 πn

Solution steps

4 cos^{2} (x) + 3 sin^{2} (x) - 4 = - tan^{2} (x)

Subtract

- tan^{2} (x)

from both sides

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

Rewrite using trig identities

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

Factor

- sin^{2} (x) + tan^{2} (x) : (tan (x) + sin (x)) (tan (x) - sin (x))

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

Solving each part separately

tan (x) + sin (x) = 0 or tan (x) - sin (x) = 0

tan (x) + sin (x) = 0 : x = 2 πn, x = π + 2 πn

tan (x) - sin (x) = 0 : x = 2 πn, x = π + 2 πn

Combine all the solutions

x = 2 πn, x = π + 2 πn

tan^{2} (x) - 2 = 3 tan (x)

3 cos (x) + sin (x) = 3

2 tan^{2} (θ) - 2 = 0

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

2 csc (x) cos (x) = csc (x)

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