## Algebra 2 Common Core

$$\tan^{2}\theta$$
Use the Pythagorean Identity: $$1+\tan^{2}\theta=\sec^{2}\theta$$ Rearrange terms: $$\sec^{2}\theta-1=\tan^{2}\theta$$ Alternatively, replace $\sec{\theta}$ with $\frac{1}{\cos\theta}$ $$\sec^{2}\theta-1= \frac{1}{cos^{2}\theta}-1$$ Common denominator: $$\sec^{2}\theta-1= \frac{1-cos^{2}\theta}{cos^{2}\theta}$$ By identity, $sin^{2}\theta+cos^{2}\theta=1$ (subtract $cos^{2}\theta$ on both sides) Therefore, replace 1-$cos^{2}\theta$ with $sin^{2}\theta$ $$\sec^{2}\theta-1=\frac{sin^{2}\theta}{cos^{2}\theta}$$ By identity: $\frac{sin\theta}{cos\theta}=tan\theta$ Hence, $$\sec^{2}\theta-1=tan^{2}\theta$$