### Home > PC3 > Chapter 11 > Lesson 11.2.2 > Problem11-106

11-106.

Solve: $\sin^4(x)-\cos^4(x)=\frac{1}{2}$

Factor.

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

Substitute using the Fundamental Pythagorean Identity.

$\sin^2(x)-\cos^2(x)=\frac{1}{2}$

Substitute using the Fundamental Pythagorean Identity, rewritten.

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

Combine like terms and subtract $1$ from both sides.

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

Divide both sides by $−2$.

$\cos^2(x)=\frac{1}{4}$

Square root both sides. Don't forget the $±$.

$\cos(x)=\pm \frac{1}{2}$

$\text{Sketch a unit circle to find all angles whose cosine is }\pm \frac{1}{2}.$