## Algebra 2 Common Core

$$\sqrt {\frac{1}{\cos^{2}\theta}-1}$$
Start with the Pythagorean Identity $$\sin^{2}\theta+\cos^{2}\theta = 1.$$ Rearrange terms and solve for $\sin\theta$: $$\sin^{2}\theta=1-\cos^{2}\theta$$ $$\sin\theta=\sqrt {1-\cos^{2}\theta}.$$ Next use the Tangent Identity and substitute in the above expression for $\sin\theta$: $$\tan\theta=\frac{\sin\theta}{\cos\theta}=\frac{\sqrt {1-\cos^{2}\theta}}{\cos\theta}=\sqrt {\frac{1}{\cos^{2}\theta}-1}.$$