The value of ` 2 sec^(2) theta - sec^(4) theta - 2 cosec^(2)theta + cosec^(4) theta` is

Options

A. `cot^(4) theta - tan^(4) theta`
B. `cot^(4) theta - tan^(2) theta`
C. `cot^(4) theta + tan^(4) theta`
D. `cos 4 theta - tan 4 theta`

Answer

To solve the expression \( 2 \sec^2 \theta - \sec^4 \theta - 2 \csc^2 \theta + \csc^4 \theta \), we can follow these steps:

### Step 1: Rearranging the Expression
We start by rearranging the terms for clarity:
\[ \csc^4 \theta - 2 \csc^2 \theta - (\sec^4 \theta - 2 \sec^2 \theta) \]

### Step 2: Factoring the Terms
Next, we can factor each part:
\[ \csc^4 \theta - 2 \csc^2 \theta = (\csc^2 \theta)^2 - 2(\csc^2 \theta) = (\csc^2 \theta - 1)^2 \]
\[ \sec^4 \theta - 2 \sec^2 \theta = (\sec^2 \theta)^2 - 2(\sec^2 \theta) = (\sec^2 \theta - 1)^2 \]

### Step 3: Substituting the Identities
Using the Pythagorean identities:
\[ \csc^2 \theta - 1 = \cot^2 \theta \]
\[ \sec^2 \theta - 1 = \tan^2 \theta \]
we can substitute these into our expression:
\[ (\cot^2 \theta)^2 - (\tan^2 \theta)^2 \]

### Step 4: Applying the Difference of Squares
Now, we can use the difference of squares formula \( a^2 - b^2 = (a - b)(a + b) \):
\[ \cot^4 \theta - \tan^4 \theta = (\cot^2 \theta - \tan^2 \theta)(\cot^2 \theta + \tan^2 \theta) \]

### Final Result
Thus, the final simplified expression is:
\[ (\cot^2 \theta - \tan^2 \theta)(\cot^2 \theta + \tan^2 \theta) \]

### Conclusion
The value of \( 2 \sec^2 \theta - \sec^4 \theta - 2 \csc^2 \theta + \csc^4 \theta \) simplifies to \( \cot^4 \theta - \tan^4 \theta \).

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