Simplify the expression. Write the final form with no fraction $$ \frac{\sin ^{2} x+2}{\cos x}+\cos x $$

Question

UpStudy AI · Accepted Answer

Step 1: Start with the given expression $$\frac{\sin^2 x + 2}{\cos x} + \cos x$$.

Step 2: Rewrite the expression by separating the terms: $$\frac{\sin^2 x}{\cos x} + \frac{2}{\cos x} + \cos x$$.

Step 3: Recognize that $$\frac{\sin^2 x}{\cos x} = 	an x \sin x$$ and $$\frac{2}{\cos x} = 2\sec x$$.

Step 4: Combine the terms: $$	an x \sin x + 2\sec x + \cos x$$.

Step 5: To eliminate the fraction, multiply the entire expression by $$\cos x$$: $$\sin^2 x + 2 + \cos^2 x$$.

Step 6: Use the Pythagorean identity $$\sin^2 x + \cos^2 x = 1$$ to simplify: $$1 + 2 = 3$$.

Step 7: The final expression is $$3$$.$$\sin^2 x + 2\sec x $$

Simplify the expression. Write the final form with no fraction \[ \frac{\sin ^{2} x+2}{\cos x}+\cos x \]