The equation in the image reads:

$\frac{\tan^2 x - 1}{\sin^2 x} = \sec^2 x - \csc x$

Let's analyze and simplify step by step.

Step 1: Simplify the numerator $\tan^2 x - 1$

Using the Pythagorean identity: $\tan^2 x = \sec^2 x - 1$ we rewrite: $\tan^2 x - 1 = \sec^2 x - 1 - 1 = \sec^2 x - 2$

So, the left-hand side becomes: $\frac{\sec^2 x - 2}{\sin^2 x}$

Step 2: Express $\sec^2 x$ and $\sin^2 x$ in terms of sine and cosine

Using the definitions: $\sec^2 x = \frac{1}{\cos^2 x}, \quad \sin^2 x = \sin^2 x,$ we rewrite the fraction: $\frac{\sec^2 x - 2}{\sin^2 x} = \frac{\frac{1}{\cos^2 x} - 2}{\sin^2 x}$

Simplify the numerator: $\frac{1}{\cos^2 x} - 2 = \frac{1 - 2\cos^2 x}{\cos^2 x}$

Thus, the fraction becomes: $\frac{\frac{1 - 2\cos^2 x}{\cos^2 x}}{\sin^2 x} = \frac{1 - 2\cos^2 x}{\cos^2 x \cdot \sin^2 x}$

Step 3: Analyze the right-hand side ($\sec^2 x - \csc x$)

Using the definitions: $\sec^2 x = \frac{1}{\cos^2 x}, \quad \csc x = \frac{1}{\sin x},$ the right-hand side becomes: $\frac{1}{\cos^2 x} - \frac{1}{\sin x}$

Combine into a single fraction: $\frac{1}{\cos^2 x} - \frac{1}{\sin x} = \frac{\sin x - \cos^2 x}{\cos^2 x \cdot \sin x}$

Step 4: Compare both sides

We now have the left-hand side: $\frac{1 - 2\cos^2 x}{\cos^2 x \cdot \sin^2 x}$ and the right-hand side: $\frac{\sin x - \cos^2 x}{\cos^2 x \cdot \sin x}$

It seems the two sides don't directly match. This suggests either an additional manipulation is required or there's an error in the original equation.

Would you like me to explore alternate steps or clarify further?

Follow-Up Questions:

1. Would you like a detailed derivation of the trigonometric identities used?
2. Should I attempt solving for $x$ under specific conditions?
3. Would you like to verify the domain or restrictions for this equation?
4. Do you want a graphing-based validation of both sides?
5. Should I explore potential simplifications or corrections for the equation?

Tip: Always verify identities by substituting key values (e.g., $x = \frac{\pi}{4}$) to check their validity.