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The given expression to evaluate is:

\[
\frac{2 \sin^2 60^\circ - \tan^2 30^\circ}{\sec^2 45^\circ}
\]

### Step-by-step solution:

1. **Evaluate the trigonometric values**:
   - \(\sin 60^\circ = \frac{\sqrt{3}}{2}\)
   - \(\tan 30^\circ = \frac{1}{\sqrt{3}}\)
   - \(\sec 45^\circ = \frac{1}{\cos 45^\circ} = \frac{1}{\frac{1}{\sqrt{2}}} = \sqrt{2}\)

2. **Square the trigonometric values**:
   - \(\sin^2 60^\circ = \left( \frac{\sqrt{3}}{2} \right)^2 = \frac{3}{4}\)
   - \(\tan^2 30^\circ = \left( \frac{1}{\sqrt{3}} \right)^2 = \frac{1}{3}\)
   - \(\sec^2 45^\circ = (\sqrt{2})^2 = 2\)

3. **Substitute these values into the expression**:
   \[
   \frac{2 \left( \frac{3}{4} \right) - \frac{1}{3}}{2}
   \]

4. **Simplify the numerator**:
   \[
   2 \times \frac{3}{4} = \frac{6}{4} = \frac{3}{2}
   \]
   So the numerator becomes:
   \[
   \frac{3}{2} - \frac{1}{3}
   \]

5. **Find the common denominator for the numerator**:
   \[
   \frac{3}{2} = \frac{9}{6}, \quad \frac{1}{3} = \frac{2}{6}
   \]
   Now, subtract:
   \[
   \frac{9}{6} - \frac{2}{6} = \frac{7}{6}
   \]

6. **Final expression**:
   \[
   \frac{\frac{7}{6}}{2} = \frac{7}{6} \times \frac{1}{2} = \frac{7}{12}
   \]

Thus, the value of the expression is:

\[
\boxed{\frac{7}{12}}
\]

Let me know if you would like further details or clarifications!

### Related Questions:
1. What is the general formula for \(\sin^2 \theta\) and \(\tan^2 \theta\)?
2. How does the identity \(\sec^2 \theta = 1 + \tan^2 \theta\) apply to this problem?
3. How can we use the Pythagorean identity to simplify trigonometric expressions?
4. What are the key values of \(\sin\), \(\tan\), and \(\sec\) for standard angles (e.g., \(30^\circ\), \(45^\circ\), \(60^\circ\))?
5. Can you express the final result \(\frac{7}{12}\) as a decimal?

**Tip**: Always check if common trigonometric identities can simplify the problem further!

Evaluate: (2 sin^2 60° - tan^2 30°) / sec^2 45°

Learn how to evaluate the expression (2 sin^2 60° - tan^2 30°) / sec^2 45°. This solution involves key trigonometric ratios such as sin, tan, and sec, as well as the Pythagorean identity. Ideal for students in Grades 10-12.

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