Alright, let's solve the equation step by step.
Given equation:
[tex]\[ 2 \sec^2 \theta - \sec^4 \theta - 2 \csc^2 \theta + \csc^4 \theta = \cot^4 \theta - \tan^4 \theta \][/tex]
We start by simplifying the expression on the left-hand side.
1. Simplify each trigonometric term:
- [tex]\(\sec^2 \theta = 1 + \tan^2 \theta\)[/tex]
- [tex]\(\sec^4 \theta = \left( \sec^2 \theta \right)^2 = (1 + \tan^2 \theta)^2\)[/tex]
- [tex]\(\csc^2 \theta = 1 + \cot^2 \theta\)[/tex]
- [tex]\(\csc^4 \theta = \left( \csc^2 \theta \right)^2 = (1 + \cot^2 \theta)^2\)[/tex]
Substituting these into the original equation:
[tex]\[ 2(1 + \tan^2 \theta) - (1 + \tan^2 \theta)^2 - 2(1 + \cot^2 \theta) + (1 + \cot^2 \theta)^2 \][/tex]
2. Distribute and simplify each term:
Expanding the terms:
[tex]\[ 2(1 + \tan^2 \theta) = 2 + 2 \tan^2 \theta \][/tex]
[tex]\[ (1 + \tan^2 \theta)^2 = 1 + 2 \tan^2 \theta + \tan^4 \theta \][/tex]
[tex]\[ 2(1 + \cot^2 \theta) = 2 + 2 \cot^2 \theta \][/tex]
[tex]\[ (1 + \cot^2 \theta)^2 = 1 + 2 \cot^2 \theta + \cot^4 \theta \][/tex]
3. Combine and simplify:
[tex]\[ 2 + 2 \tan^2 \theta - (1 + 2 \tan^2 \theta + \tan^4 \theta) - 2 - 2 \cot^2 \theta + (1 + 2 \cot^2 \theta + \cot^4 \theta) \][/tex]
Combine like terms:
[tex]\[ = 2 + 2 \tan^2 \theta - 1 - 2 \tan^2 \theta - \tan^4 \theta - 2 - 2 \cot^2 \theta + 1 + 2 \cot^2 \theta + \cot^4 \theta \][/tex]
4. Simplify the final result:
Terms cancel out:
[tex]\[ 2 - 2 + \cot^4 \theta - \tan^4 \theta \][/tex]
[tex]\[ = \cot^4 \theta - \tan^4 \theta \][/tex]
[tex]\[ = 2 \tan^2 \theta - 2 (\cot^2 \theta + 1) (\tan^2 \theta + 1)^2 - (\tan^2 \theta + 1)^2 + 2 \][/tex]
[tex]\[ = 2 \tan^2 \theta - 2 (1 + \tan^2 \theta)^2 - 2 (1 + \cot^2 \theta) + (1 + \cot^2 \theta)^2 \][/tex]
[tex]\[ = 2 \tan^2 \theta - 2 (\cot^2 \theta + 1) (\tan^2 \theta + 1)^2 - (\tan^2 \theta + 1)^2 + 2 \][/tex]
So, the simplified form of the given equation is:
[tex]\[ 2 \tan^2 \theta - 2 (\cot^2 \theta + 1) (\tan^2 \theta + 1)^2 - (\tan^2 \theta + 1)^2 + 2 \][/tex]
This is the detailed step-by-step solution for the given trigonometric identity.
