CPM Homework Help : PCT Problem 8-17

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

Step 1:

Factor.

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

Step 2:

Substitute using the Fundamental Pythagorean Identity.

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

Step 3:

Substitute using the Fundamental Pythagorean Identity, rewritten.

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

Step 4:

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

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

Step 5:

Divide both sides by $−2$ .

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

Step 6:

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

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

Step 7:

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