Sure, let's solve the equation [tex]\(1 - \sin \theta \cdot \cos \theta \cdot \tan \theta = \cos^2 \theta\)[/tex] step by step.
1. Start with the given equation:
[tex]\[
1 - \sin \theta \cdot \cos \theta \cdot \tan \theta = \cos^2 \theta
\][/tex]
2. Use the definition of the tangent function:
[tex]\[
\tan \theta = \frac{\sin \theta}{\cos \theta}
\][/tex]
3. Substitute the definition of [tex]\(\tan \theta\)[/tex] into the equation:
[tex]\[
1 - \sin \theta \cdot \cos \theta \cdot \frac{\sin \theta}{\cos \theta} = \cos^2 \theta
\][/tex]
4. Simplify the expression:
[tex]\[
1 - \sin \theta \cdot \sin \theta = \cos^2 \theta
\][/tex]
5. Combine the [tex]\(\sin^2 \theta\)[/tex] terms:
[tex]\[
1 - \sin^2 \theta = \cos^2 \theta
\][/tex]
6. Recall the Pythagorean identity:
[tex]\[
\sin^2 \theta + \cos^2 \theta = 1
\][/tex]
7. Using the Pythagorean identity, rewrite [tex]\(1 - \sin^2 \theta\)[/tex]:
[tex]\[
1 - \sin^2 \theta = \cos^2 \theta
\][/tex]
8. Thus, the equation simplifies to:
[tex]\[
\cos^2 \theta = \cos^2 \theta
\][/tex]
We've shown through these steps that:
[tex]\[
1 - \sin \theta \cdot \cos \theta \cdot \tan \theta = \cos^2 \theta
\][/tex]
holds true for all [tex]\(\theta\)[/tex].
