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

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

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3/(cos^2(x))=3+2tan^2(x)

\frac{3}{cos ^{2} ( x )} = 3 + 2 tan^{2} (x)

x = 2 πn, x = π + 2 πn

Solution steps

\frac{3}{cos ^{2} ( x )} = 3 + 2 tan^{2} (x)

Subtract

3 + 2 tan^{2} (x)

from both sides

\frac{3}{cos ^{2} ( x )} - 3 - 2 tan^{2} (x) = 0

Simplify

\frac{3}{cos ^{2} ( x )} - 3 - 2 tan^{2} (x) : \frac{3 - 3 cos ^{2} ( x ) - 2 tan ^{2} ( x ) cos ^{2} ( x )}{cos ^{2} ( x )}

\frac{3 - 3 cos ^{2} ( x ) - 2 tan ^{2} ( x ) cos ^{2} ( x )}{cos ^{2} ( x )} = 0

\frac{f ( x )}{g ( x )} = 0 \Rightarrow f (x) = 0

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

Express with sin, cos

3 - 3 cos^{2} (x) - 2 (\frac{sin ( x )}{cos ( x )})^{2} cos^{2} (x) = 0

Simplify

3 - 3 cos^{2} (x) - 2 (\frac{sin ( x )}{cos ( x )})^{2} cos^{2} (x) : 3 - 3 cos^{2} (x) - 2 sin^{2} (x)

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

Add

2 sin^{2} (x)

to both sides

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

Square both sides

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

Subtract

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

from both sides

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

Factor

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

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

Solving each part separately

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

3 - 3 cos^{2} (x) + 2 sin^{2} (x) = 0 : x = 2 πn, x = π + 2 πn

3 - 3 cos^{2} (x) - 2 sin^{2} (x) = 0 : x = 2 πn, x = π + 2 πn

Combine all the solutions

x = 2 πn, x = π + 2 πn

Verify solutions by plugging them into the original equation

x = 2 πn, x = π + 2 πn

13 \cdot sin (x) = 12

11 sin (θ) = 11 cos (θ)

1 - 2 tan (x) = 0

sin (a) = 0.747

1 - tan^{2} (y) = 2 sec (y) - 4

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