To simplify the expression \(\frac{3 - 4\sin^2 \theta}{\cos^2 \theta} + 2\tan^2 \theta\), we start by rewriting \(\sin^2 \theta\) and \(\tan^2 \theta\) using trigonometric identities:

• \(\sin^2 \theta = 1 - \cos^2 \theta\)
• \(\tan^2 \theta = \frac{\sin^2 \theta}{\cos^2 \theta}\)

Substituting these identities into the expression:

\(\frac{3 - 4(1 - \cos^2 \theta)}{\cos^2 \theta} + 2\left(\frac{1 - \cos^2 \theta}{\cos^2 \theta}\right)\)

Simplify inside the brackets:

\(\frac{3 - 4 + 4\cos^2 \theta}{\cos^2 \theta} + \frac{2 - 2\cos^2 \theta}{\cos^2 \theta}\)

This reduces to:

\(\frac{-1 + 4\cos^2 \theta}{\cos^2 \theta} + \frac{2 - 2\cos^2 \theta}{\cos^2 \theta}\)

Separate the fractions:

\(\frac{-1}{\cos^2 \theta} + 4 + \frac{2}{\cos^2 \theta} - 2\)

Combine like terms:

\((4 - 2) + \left(\frac{-1 + 2}{\cos^2 \theta}\right)\)

Simplify further:

\(2 + \frac{1}{\cos^2 \theta}\)

Recognize that \(\frac{1}{\cos^2 \theta} = \sec^2 \theta = 1 + \tan^2 \theta\)

Thus:

\(2 + (1 + \tan^2 \theta)\)

Simplifying gives:

\(3 + \tan^2 \theta\)

Therefore, the simplified expression is \(3 + \tan^2 \theta\).