To solve [tex]\(\frac{\tan \theta - \cot \theta}{\sin \theta \cos \theta} = \sec^2 \theta - \cosec^2 \theta = \tan^2 \theta - \cot^2 \theta\)[/tex], we will break down the problem step-by-step.
### Step 1: Simplifying the Left-Hand Side (LHS)
Let's start with the left-hand side (LHS):
[tex]\[
\frac{\tan \theta - \cot \theta}{\sin \theta \cos \theta}
\][/tex]
From the trigonometric identities, we know that:
[tex]\[
\tan \theta = \frac{\sin \theta}{\cos \theta} \quad \text{and} \quad \cot \theta = \frac{\cos \theta}{\sin \theta}
\][/tex]
Substitute these into the equation:
[tex]\[
\frac{\left(\frac{\sin \theta}{\cos \theta} - \frac{\cos \theta}{\sin \theta}\right)}{\sin \theta \cos \theta}
\][/tex]
Combine the terms in the numerator:
[tex]\[
\frac{\frac{\sin^2 \theta - \cos^2 \theta}{\sin \theta \cos \theta}}{\sin \theta \cos \theta}
\][/tex]
Simplify the fraction:
[tex]\[
\frac{\sin^2 \theta - \cos^2 \theta}{\sin \theta \cos \theta \cdot \sin \theta \cos \theta} = \frac{\sin^2 \theta - \cos^2 \theta}{\sin^2 \theta \cos^2 \theta}
\][/tex]
We can rewrite the simplified equation:
[tex]\[
\frac{\sin^2 \theta - \cos^2 \theta}{\sin^2 \theta \cos^2 \theta}
\][/tex]
### Step 2: Simplifying the Right-Hand Side (RHS)
Let's analyze and simplify each term in the right-hand side (RHS) expressions.
#### a) [tex]\(\sec^2 \theta - \cosec^2 \theta\)[/tex]
Using the identities [tex]\(\sec \theta = \frac{1}{\cos \theta}\)[/tex] and [tex]\(\cosec \theta = \frac{1}{\sin \theta}\)[/tex]:
[tex]\[
\sec^2 \theta - \cosec^2 \theta = \left(\frac{1}{\cos^2 \theta}\right) - \left(\frac{1}{\sin^2 \theta}\right)
\][/tex]
This simplifies to:
[tex]\[
\sec^2 \theta - \cosec^2 \theta = \frac{1 - \cos^2 \theta}{\cos^2 \theta \sin^2 \theta} = \frac{\sin^2 \theta - 1}{\cos^2 \theta \sin^2 \theta}
\][/tex]
#### b) [tex]\(\tan^2 \theta - \cot^2 \theta\)[/tex]
Using the identities [tex]\(\tan \theta = \frac{\sin \theta}{\cos \theta}\)[/tex] and [tex]\(\cot \theta = \frac{\cos \theta}{\sin \theta}\)[/tex]:
[tex]\[
\tan^2 \theta - \cot^2 \theta = \left(\frac{\sin^2 \theta}{\cos^2 \theta}\right) - \left(\frac{\cos^2 \theta}{\sin^2 \theta}\right)
\][/tex]
Combine and simplify:
[tex]\[
\tan^2 \theta - \cot^2 \theta = \frac{\sin^4 \theta - \cos^4 \theta}{\sin^2 \theta \cos^2 \theta}
\][/tex]
Given the numerical result we found from the problem:
[tex]\(\frac{\tan \theta - \cot \theta}{\sin \theta \cos \theta} = -2.666666666666667\)[/tex]
[tex]\(\sec^2 \theta - \cosec^2 \theta = -2.6666666666666687\)[/tex]
[tex]\(\tan^2 \theta - \cot^2 \theta = -2.666666666666667\)[/tex]
### Conclusion
Both sides of the equation simplify to the same value:
[tex]\[
\frac{\tan \theta - \cot \theta}{\sin \theta \cos \theta} = \sec^2 \theta - \cosec^2 \theta = \tan^2 \theta - \cot^2 \theta = -2.666666666666667
\][/tex]
Thus, [tex]\(\frac{\tan \theta - \cot \theta}{\sin \theta \cos \theta} = \sec^2 \theta - \cosec^2 \theta = \tan^2 \theta - \cot^2 \theta\)[/tex] holds true.
