To find the expression [tex]\(\tan^3 \theta - \cot^3 \theta\)[/tex], we can proceed as follows:
1. Understand the individual components:
- [tex]\(\tan^3 \theta\)[/tex] is the tangent of [tex]\(\theta\)[/tex] raised to the power of 3.
- [tex]\(\cot^3 \theta\)[/tex] is the cotangent of [tex]\(\theta\)[/tex] raised to the power of 3. Recall that cotangent is the reciprocal of the tangent function, i.e., [tex]\(\cot \theta = \frac{1}{\tan \theta}\)[/tex].
2. Express [tex]\(\cot^3 \theta\)[/tex]in terms of [tex]\(\tan \theta\)[/tex]:
[tex]\[
\cot^3 \theta = \left( \frac{1}{\tan \theta} \right)^3 = \frac{1}{\tan^3 \theta}
\][/tex]
3. Substitute back into the original expression:
[tex]\[
\tan^3 \theta - \cot^3 \theta = \tan^3 \theta - \frac{1}{\tan^3 \theta}
\][/tex]
Therefore, [tex]\(\tan^3 \theta - \cot^3 \theta\)[/tex] simplifies to:
[tex]\[
\tan^3 \theta - \frac{1}{\tan^3 \theta}
\][/tex]
