# How to simplify cos^2 2theta-sin^2 2theta?

Jun 21, 2018

$\cos 4 \theta$

#### Explanation:

${\cos}^{2} \left(2 \theta\right) - {\sin}^{2} \left(2 \theta\right)$

=$\cos 2 \left(2 \theta\right)$

=$\cos 4 \theta$

This is basically the double angle formula

Recall that $\cos 2 x = {\cos}^{2} x - {\sin}^{2} x$

Now replace $x$ with $2 \theta$

$\cos 2 \left(2 \theta\right) = {\cos}^{2} \left(2 \theta\right) - {\sin}^{2} \left(2 \theta\right)$

$\cos 4 \theta = {\cos}^{2} 2 \theta - {\sin}^{2} 2 \theta$

Jun 21, 2018

This simplifies to $\cos \left(4 \theta\right)$

#### Explanation:

Let $A = 2 \theta$.

Then the expression becomes ${\cos}^{2} A - {\sin}^{2} A = \cos \left(2 A\right)$. Now reverse the substitution to see that the expression gives $\cos \left(2 \left(2 \theta\right)\right) = \cos \left(4 \theta\right)$

Hopefully this helps!