Prove the following identities: `2sec^2theta-sec^4theta-2cos e c^2theta+cosec^4 theta=cot^4theta-tan^4theta`

Explanation

To prove the identity \( 2\sec^2\theta - \sec^4\theta - 2\cos^2\theta + \csc^4\theta = \cot^4\theta - \tan^4\theta \), we will start with the left-hand side (LHS) and manipulate it to show that it equals the right-hand side (RHS).

### Step 1: Rewrite the LHS
We start with the LHS:
\[ LHS = 2\sec^2\theta - \sec^4\theta - 2\cos^2\theta + \csc^4\theta \]
Using the identities \( \sec^2\theta = \frac{1}{\cos^2\theta} \) and \( \csc^2\theta = \frac{1}{\sin^2\theta} \), we can express \( \csc^4\theta \) as \( \left(\csc^2\theta\right)^2 = \left(\frac{1}{\sin^2\theta}\right)^2 = \frac{1}{\sin^4\theta} \).
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