Sure, let's establish the trigonometric identity:
[tex]\[
(1 + \tan^2 \theta) \cos^2 \theta = 1
\][/tex]
1. Start with the left-hand side (LHS) of the equation:
[tex]\[
\text{LHS} = (1 + \tan^2 \theta) \cos^2 \theta
\][/tex]
2. Recall the Pythagorean identity involving tangent and secant:
[tex]\[
1 + \tan^2 \theta = \sec^2 \theta
\][/tex]
3. Substitute [tex]\(\sec^2 \theta\)[/tex] in place of [tex]\(1 + \tan^2 \theta\)[/tex]:
[tex]\[
\text{LHS} = \sec^2 \theta \cdot \cos^2 \theta
\][/tex]
4. Recall the definition of secant. Since [tex]\(\sec \theta = \frac{1}{\cos \theta}\)[/tex], we have:
[tex]\[
\sec^2 \theta = \frac{1}{\cos^2 \theta}
\][/tex]
5. Substitute [tex]\(\frac{1}{\cos^2 \theta}\)[/tex] for [tex]\(\sec^2 \theta\)[/tex]:
[tex]\[
\text{LHS} = \frac{1}{\cos^2 \theta} \cdot \cos^2 \theta
\][/tex]
6. Simplify the expression:
[tex]\[
\frac{1}{\cos^2 \theta} \cdot \cos^2 \theta = 1
\][/tex]
Thus, we have shown that:
[tex]\[
(1 + \tan^2 \theta) \cos^2 \theta = 1
\][/tex]
Therefore, the given identity is established.
