saini0himanshu
Answered

[tex]\[
2 \sec^2 \theta - \sec^4 \theta - 2 \operatorname{cosec}^2 \theta + \operatorname{cosec}^4 \theta = \cot^4 \theta - \tan^4 \theta
\][/tex]

Simplify the expression:
[tex]\[
2 \left(1 + \tan^2 \theta\right) - \left(1 + \tan^2 \theta\right)^2 - 2 \left(1 + \cot^2 \theta\right) - \left(1 + \cot^2 \theta\right)^2
\][/tex]



Answer :

GinnyAnswer
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.

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