Sure, let's break down the calculation step-by-step for the expression [tex]\(\cot^2(2\beta) - \csc^2(2\beta)\)[/tex].
1. Understand Cotangent and Cosecant Functions:
- [tex]\(\cot(x) = \frac{1}{\tan(x)}\)[/tex]
- [tex]\(\csc(x) = \frac{1}{\sin(x)}\)[/tex]
2. Express Functions in Terms of [tex]\(2\beta\)[/tex]:
- [tex]\(\cot(2\beta) = \frac{1}{\tan(2\beta)}\)[/tex]
- [tex]\(\csc(2\beta) = \frac{1}{\sin(2\beta)}\)[/tex]
3. Square Each Function:
- [tex]\(\cot^2(2\beta) = \left(\frac{1}{\tan(2\beta)}\right)^2 = \frac{1}{\tan^2(2\beta)}\)[/tex]
- [tex]\(\csc^2(2\beta) = \left(\frac{1}{\sin(2\beta)}\right)^2 = \frac{1}{\sin^2(2\beta)}\)[/tex]
4. Substitute and Simplify the Expression:
- [tex]\(\cot^2(2\beta) - \csc^2(2\beta)\)[/tex]: Let's substitute the above results into this expression.
- [tex]\[
\cot^2(2\beta) - \csc^2(2\beta) = \frac{1}{\tan^2(2\beta)} - \frac{1}{\sin^2(2\beta)}
\][/tex]
Thus, the result of [tex]\(\cot^2(2\beta) - \csc^2(2\beta)\)[/tex] is:
[tex]\[
\frac{1}{\tan^2(2\beta)} - \frac{1}{\sin^2(2\beta)}
\][/tex]
By following these steps, we reach our final result, [tex]\(\cot^2(2\beta) - \csc^2(2\beta) = \frac{1}{\tan^2(2\beta)} - \frac{1}{\sin^2(2\beta)}\)[/tex].
