Certainly! Let's demonstrate that the trigonometric identity [tex]\(\cos^2 \theta + \sin^2 \theta = 1\)[/tex] holds true.
1. Consider the expressions for cosine and sine squared:
We start with the expressions [tex]\(\cos^2 \theta\)[/tex] and [tex]\(\sin^2 \theta\)[/tex].
2. Combine the expressions:
Add these two expressions together:
[tex]\[
\cos^2 \theta + \sin^2 \theta
\][/tex]
3. Recognize the fundamental trigonometric identity:
From basic trigonometric principles, we know that the Pythagorean identity states:
[tex]\[
\sin^2 \theta + \cos^2 \theta = 1
\][/tex]
4. Simplify the combined expression:
Based on this fundamental identity, it is already established that:
[tex]\[
\sin^2 \theta + \cos^2 \theta = 1
\][/tex]
Therefore, it follows that the combination of [tex]\(\cos^2 \theta\)[/tex] and [tex]\(\sin^2 \theta\)[/tex] simplifies to 1. So:
[tex]\[
\cos^2 \theta + \sin^2 \theta = 1
\][/tex]
The trigonometric identity [tex]\(\cos^2 \theta + \sin^2 \theta = 1\)[/tex] is thus confirmed to be true.
