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

The general solution for 2sin^2(x)=sec(x) is No Solution for x\in\mathbb{R}

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2sin^2(x)=sec(x)

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

No Solution for x \in R

Solution steps

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

Square both sides

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

Subtract

sec^{2} (x)

from both sides

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

Express with sin, cos

\frac{- 1 + 4 cos ^{2} ( x ) sin ^{4} ( x )}{cos ^{2} ( x )} = 0

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

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

Factor

- 1 + 4 cos^{2} (x) sin^{4} (x) : (2 sin^{2} (x) cos (x) + 1) (2 sin^{2} (x) cos (x) - 1)

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

Solving each part separately

2 sin^{2} (x) cos (x) + 1 = 0 or 2 sin^{2} (x) cos (x) - 1 = 0

2 sin^{2} (x) cos (x) + 1 = 0 :

No Solution

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

No Solution

Combine all the solutions

No Solution

Verify solutions by plugging them into the original equation

No Solution for x \in R

tan (x) = \frac{12}{6}

3 sin (θ) - 2 sin (θ) cos (θ) = 0

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

13 \cdot sin (x) = 12

11 sin (θ) = 11 cos (θ)

What is the general solution for 2sin^2(x)=sec(x) ?
The general solution for 2sin^2(x)=sec(x) is No Solution for x\in\mathbb{R}