Simplifying the Trigonometric Expression

The problem asks to simplify the given trigonometric expression:

\(\frac{\sec^2 x + \tan^2 x}{\sin^2 x + 1}\)

Step-by-Step Simplification

We use fundamental trigonometric identities to simplify the expression.

1. Recall the Pythagorean identity: \(\sec^2 x = 1 + \tan^2 x\). Substitute this into the numerator:

   Numerator = \(\sec^2 x + \tan^2 x = (1 + \tan^2 x) + \tan^2 x = 1 + 2\tan^2 x\).

2. The expression now becomes:

   \(\frac{1 + 2\tan^2 x}{\sin^2 x + 1}\)

3. Use the identity \(\tan x = \frac{\sin x}{\cos x}\), which implies \(\tan^2 x = \frac{\sin^2 x}{\cos^2 x}\). Substitute this into the numerator:

   Numerator = \(1 + 2 \left( \frac{\sin^2 x}{\cos^2 x} \right) = \frac{\cos^2 x + 2\sin^2 x}{\cos^2 x}\).

4. Simplify the numerator further using \(\cos^2 x + \sin^2 x = 1\):

   Numerator = \(\frac{(\cos^2 x + \sin^2 x) + \sin^2 x}{\cos^2 x} = \frac{1 + \sin^2 x}{\cos^2 x}\).

5. Substitute this back into the expression:

   \(\frac{\frac{1 + \sin^2 x}{\cos^2 x}}{\sin^2 x + 1}\)

6. Simplify the compound fraction:

   \(\frac{1 + \sin^2 x}{\cos^2 x (\sin^2 x + 1)}\)

7. Cancel the common term \((1 + \sin^2 x)\) from the numerator and denominator:

   \(\frac{1}{\cos^2 x}\)

Alternatively, we know that \(\sec x = \frac{1}{\cos x}\), so \(\sec^2 x = \frac{1}{\cos^2 x}\). The simplified expression is \(\frac{1}{\cos^2 x}\).