Find All Complex Solutions sin(x)^2+5cos(x/2)^2=2

$\sin^{2} (x) + 5 \cos^{2} (\frac{x}{2}) = 2$

Step 1

Subtract $2$ from both sides of the equation.

$\sin^{2} (x) + 5 \cos^{2} (\frac{x}{2}) - 2 = 0$

Step 2

Replace $\sin^{2} (x)$ with $1 - \cos^{2} (x)$ .

$(1 - \cos^{2} (x)) + 5 \cos^{2} (\frac{x}{2}) - 2 = 0$

Step 3

Simplify the left side.

Apply pythagorean identity.

$\sin^{2} (x) + 5 \cos^{2} (\frac{x}{2}) - 2 = 0$

Step 4

Replace $\sin^{2} (x)$ with $1 - \cos^{2} (x)$ .

$(1 - \cos^{2} (x)) + 5 \cos^{2} (\frac{x}{2}) - 2 = 0$

Step 5

Apply pythagorean identity.

$\sin^{2} (x) + 5 \cos^{2} (\frac{x}{2}) - 2 = 0$

Step 6

Replace $\sin^{2} (x)$ with $1 - \cos^{2} (x)$ .

$(1 - \cos^{2} (x)) + 5 \cos^{2} (\frac{x}{2}) - 2 = 0$