To simplify the expression [tex]\(\sin(A) \tan(A) + \cos(A)\)[/tex] into an expression involving only [tex]\(\sin(A)\)[/tex] or [tex]\(\cos(A)\)[/tex], we can follow these steps:
1. Recall the trigonometric identity for [tex]\(\tan(A)\)[/tex]:
[tex]\[
\tan(A) = \frac{\sin(A)}{\cos(A)}
\][/tex]
2. Substitute [tex]\(\tan(A)\)[/tex] into the given expression:
[tex]\[
\sin(A) \tan(A) + \cos(A) = \sin(A) \left( \frac{\sin(A)}{\cos(A)} \right) + \cos(A)
\][/tex]
3. Simplify the first term:
[tex]\[
\sin(A) \left( \frac{\sin(A)}{\cos(A)} \right) = \frac{\sin^2(A)}{\cos(A)}
\][/tex]
4. So the expression now is:
[tex]\[
\frac{\sin^2(A)}{\cos(A)} + \cos(A)
\][/tex]
5. To combine these terms, we need a common denominator:
[tex]\[
\frac{\sin^2(A)}{\cos(A)} + \frac{\cos^2(A)}{\cos(A)}
\][/tex]
6. Combine the fractions:
[tex]\[
\frac{\sin^2(A) + \cos^2(A)}{\cos(A)}
\][/tex]
7. Use the Pythagorean identity, which states that:
[tex]\[
\sin^2(A) + \cos^2(A) = 1
\][/tex]
8. Substitute [tex]\(1\)[/tex] for [tex]\(\sin^2(A) + \cos^2(A)\)[/tex] in the expression:
[tex]\[
\frac{1}{\cos(A)}
\][/tex]
Thus, the simplified expression is:
[tex]\[
\frac{1}{\cos(A)}
\][/tex]
