# Please help me simplify. What is cos^4theta -sin^4theta +sin^2theta equal to?

May 31, 2018

${\cos}^{2} \left(x\right)$

#### Explanation:

We have
(cos^4(x)-sin^4(x))=(cos^2(x)-sin^2(x)(cos^2(x)+sin^2(x))+sin^2(x)
Now we use that
${\sin}^{2} \left(x\right) + {\cos}^{2} \left(x\right) = 1$

So we get
${\cos}^{2} \left(x\right) - {\sin}^{2} \left(x\right) + {\sin}^{2} \left(x\right) = {\cos}^{2} \left(x\right)$

May 31, 2018

${\cos}^{2} \theta$

#### Explanation:

${\cos}^{4} \theta - {\sin}^{4} \theta \text{ is a "color(blue)"difference of squares}$

â€¢color(white)(x)a^2-b^2=(a-b)(a+b)

$\text{here "a=cos^2theta" and } b = {\sin}^{2} \theta$

${\cos}^{4} \theta - {\sin}^{4} \theta = \left({\cos}^{2} \theta - {\sin}^{2} \theta\right) \left({\cos}^{2} \theta + {\sin}^{2} \theta\right)$

$\left[{\cos}^{2} \theta + {\sin}^{2} \theta = 1\right]$

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

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

$= {\cos}^{2} \theta$