# How do you simplify \frac{\sin^4 \theta - \cos^4 \theta}{\sin^2 \theta - \cos^2 \theta}  using the trigonometric identities?

Dec 22, 2014

You should get: $\frac{{\sin}^{4} \left(\theta\right) - {\cos}^{4} \left(\theta\right)}{{\sin}^{2} \left(\theta\right) - {\cos}^{2} \left(\theta\right)} = 1$

This is because in the numerator you have:

${\sin}^{4} \left(\theta\right) - {\cos}^{4} \left(\theta\right)$ which is the same as:

$\left[{\sin}^{2} \left(\theta\right) + {\cos}^{2} \left(\theta\right)\right] \cdot \left[{\sin}^{2} \left(\theta\right) - {\cos}^{2} \left(\theta\right)\right]$

(Remember that $\left(a + b\right) \cdot \left(a - b\right) = {a}^{2} - {b}^{2}$)

$\frac{\left[{\sin}^{2} \left(\theta\right) + {\cos}^{2} \left(\theta\right)\right] \cdot \left[{\sin}^{2} \left(\theta\right) - {\cos}^{2} \left(\theta\right)\right]}{{\sin}^{2} \left(\theta\right) - {\cos}^{2} \left(\theta\right)} =$
You can now simplyfy the two: ${\sin}^{2} \left(\theta\right) - {\cos}^{2} \left(\theta\right)$ in the numerator and denominator.
$\left[{\sin}^{2} \left(\theta\right) + {\cos}^{2} \left(\theta\right)\right]$ which is always equal to 1 (for every angle $\theta$)