Certainly! To verify the identity [tex]\(\tan^2 \theta - \cot^2 \theta = \sec^2 \theta (1 - \cot^2 \theta)\)[/tex], let's carefully go through the expressions step by step.
1. Left-hand side (LHS):
[tex]\[
\tan^2 \theta - \cot^2 \theta
\][/tex]
We know that [tex]\(\cot \theta = \frac{1}{\tan \theta}\)[/tex], thus:
[tex]\[
\cot^2 \theta = \left( \frac{1}{\tan \theta} \right)^2 = \frac{1}{\tan^2 \theta}
\][/tex]
Therefore, substituting [tex]\(\cot^2 \theta\)[/tex] in the LHS:
[tex]\[
\tan^2 \theta - \frac{1}{\tan^2 \theta}
\][/tex]
2. Right-hand side (RHS):
[tex]\[
\sec^2 \theta (1 - \cot^2 \theta)
\][/tex]
We know that [tex]\(\sec \theta = \frac{1}{\cos \theta}\)[/tex] and using [tex]\(\cot \theta = \frac{\cos \theta}{\sin \theta}\)[/tex]:
[tex]\[
\sec^2 \theta = 1 + \tan^2 \theta
\][/tex]
Therefore, substituting [tex]\(\sec^2 \theta\)[/tex] and simplifying [tex]\(1 - \cot^2 \theta\)[/tex]:
[tex]\[
\sec^2 \theta \left(1 - \frac{1}{\tan^2 \theta}\right)
\][/tex]
3. Simplifying inside the parentheses on the RHS:
[tex]\[
1 - \frac{1}{\tan^2 \theta} = \frac{\tan^2 \theta - 1}{\tan^2 \theta}
\][/tex]
Thus, the RHS expression becomes:
[tex]\[
\sec^2 \theta \cdot \frac{\tan^2 \theta - 1}{\tan^2 \theta}
\][/tex]
We know:
[tex]\[
\sec^2 \theta = 1 + \tan^2 \theta
\][/tex]
Substituting [tex]\(\sec^2 \theta\)[/tex] back in:
[tex]\[
(1 + \tan^2 \theta) \cdot \frac{\tan^2 \theta - 1}{\tan^2 \theta}
\][/tex]
Rewriting the RHS fully:
[tex]\[
(1 + \tan^2 \theta) \cdot \frac{\tan^2 \theta - 1}{\tan^2 \theta}
\][/tex]
We get the product:
[tex]\[
\frac{(\tan^2 \theta - 1)(1 + \tan^2 \theta)}{\tan^2 \theta}
\][/tex]
After performing these steps and comparing the simplified forms, we see that:
- The LHS simplifies to [tex]\(\tan^2 \theta - \frac{1}{\tan^2 \theta}\)[/tex]
- The RHS simplifies to [tex]\(\frac{(\tan^2 \theta - 1)(1 + \tan^2 \theta)}{\tan^2 \theta}\)[/tex]
Upon cross-comparison, it is clear that [tex]\(\tan^2 \theta - \cot^2 \theta\)[/tex] and [tex]\(\sec^2 \theta (1 - \cot^2 \theta)\)[/tex] are not identical expressions for the given function properties. Therefore, the original equation:
[tex]\[
\tan^2 \theta - \cot^2 \theta = \sec^2 \theta (1 - \cot^2 \theta)
\][/tex]
is not valid for all [tex]\(\theta\)[/tex].
Hence, we conclude:
The statement [tex]\(\tan^2 \theta - \cot^2 \theta = \sec^2 \theta (1 - \cot^2 \theta)\)[/tex] is False.
