Questions: Simplify completely into an expression with sin(A) or cos(A) only. You will need to write cos(A) as (cos(A))/1 and add the two fractions. sin(A) tan(A)+cos(A)=

$Simplify completely into an expression with sin(A) or cos(A) only. You will need to write cos(A) as (cos(A))/1 and add the two fractions. sin(A) tan(A)+cos(A)=$

Transcript text: Simplify completely into an expression with $\sin (A)$ or $\cos (A)$ only. You will need to write $\cos (A)$ as $\frac{\cos (A)}{1}$ and add the two fractions. \[ \sin (A) \tan (A)+\cos (A)= \]

Solution

Solution Steps

To simplify the given expression, we need to express everything in terms of either $\sin(A)$ or $\cos(A)$. We know that $\tan(A) = \frac{\sin(A)}{\cos(A)}$. Substituting this into the expression and then combining the terms will give us the simplified form.

Step 1: Rewrite the Expression

We start with the expression: \[ \sin(A) \tan(A) + \cos(A) \] Substituting $\tan(A) = \frac{\sin(A)}{\cos(A)}$, we rewrite the expression as: \[ \sin(A) \cdot \frac{\sin(A)}{\cos(A)} + \cos(A) \]

Step 2: Combine the Terms

This simplifies to: \[ \frac{\sin^2(A)}{\cos(A)} + \cos(A) \] To combine these terms, we express $\cos(A)$ as $\frac{\cos^2(A)}{\cos(A)}$: \[ \frac{\sin^2(A)}{\cos(A)} + \frac{\cos^2(A)}{\cos(A)} = \frac{\sin^2(A) + \cos^2(A)}{\cos(A)} \]

Step 3: Apply the Pythagorean Identity

Using the Pythagorean identity $\sin^2(A) + \cos^2(A) = 1$, we can simplify the expression further: \[ \frac{1}{\cos(A)} \]