To verify the trigonometric identity [tex]\(\left(1 - \cos^2 A\right)\left(1 + \tan^2 A\right) = \tan^2 A\)[/tex], we will use some well-known trigonometric identities. Here is the detailed step-by-step verification:
1. Pythagorean Identity:
[tex]\[
\sin^2 A + \cos^2 A = 1
\][/tex]
Rearranging this identity, we get:
[tex]\[
\sin^2 A = 1 - \cos^2 A
\][/tex]
2. Substituting [tex]\(\sin^2 A\)[/tex] for [tex]\(1 - \cos^2 A\)[/tex]:
[tex]\[
1 - \cos^2 A = \sin^2 A
\][/tex]
So the left-hand side of the equation becomes:
[tex]\[
(\sin^2 A)(1 + \tan^2 A)
\][/tex]
3. Identity involving [tex]\(\sec^2 A\)[/tex]:
Another important trigonometric identity is:
[tex]\[
1 + \tan^2 A = \sec^2 A
\][/tex]
where [tex]\(\sec A = \frac{1}{\cos A}\)[/tex].
4. Substituting [tex]\(\sec^2 A\)[/tex] for [tex]\(1 + \tan^2 A\)[/tex]:
[tex]\[
1 + \tan^2 A = \sec^2 A
\][/tex]
Thus, the expression now is:
[tex]\[
(\sin^2 A)(\sec^2 A)
\][/tex]
5. Rewrite [tex]\(\sec^2 A\)[/tex] in terms of [tex]\( \cos A \)[/tex]:
[tex]\[
\sec^2 A = \frac{1}{\cos^2 A}
\][/tex]
Substituting this into the expression gives:
[tex]\[
(\sin^2 A) \left(\frac{1}{\cos^2 A}\right)
\][/tex]
6. Simplify the expression:
[tex]\[
\frac{\sin^2 A}{\cos^2 A}
\][/tex]
7. Recognize the resulting expression:
[tex]\[
\frac{\sin^2 A}{\cos^2 A} = \tan^2 A
\][/tex]
Thus, we have verified that the left-hand side is equal to the right-hand side of the given equation:
[tex]\[
\left(1 - \cos^2 A\right)\left(1 + \tan^2 A\right) = \tan^2 A
\][/tex]
Therefore, the trigonometric identity [tex]\(\left(1 - \cos^2 A\right)\left(1 + \tan^2 A\right) = \tan^2 A\)[/tex] holds true.
