## Algebra 2 Common Core

$$\sec\theta$$
Simplify $$\cos\theta+\sin\theta\tan\theta$$ by substituting the Tangent Identity: $$\tan\theta=\frac{\sin\theta}{\cos\theta}$$ to obtain $$\cos\theta+\sin\theta\tan\theta=\cos\theta+\sin\theta\bigg(\frac{\sin\theta}{\cos\theta}\bigg)=\cos\theta+\frac{\sin^{2}\theta}{\cos\theta}$$ Now multiply both the numerator and the denominator of the first term by $\cos\theta$ $$=\cos\theta\bigg(\frac{\cos\theta}{\cos\theta}\bigg)+\frac{\sin^{2}\theta}{\cos\theta}$$ $$=\frac{\cos^{2}\theta+\sin^{2}\theta}{\cos\theta}$$ Next, use the Pythagorean Identity: $$\sin^{2}\theta+\cos^{2}\theta=1$$ and finally the Reciprocal Identity: $$\sec\theta=\frac{1}{\cos\theta}$$ to obtain $$\cos\theta+\sin\theta\tan\theta=\frac{1}{\cos\theta}=\sec\theta$$