Evaluation of the Expression
To evaluate the expression \(2\tan^2 45^\circ + \cos^2 30^\circ - \sin^2 60^\circ\), we can break it down into its components.
Step 1: Calculate \( \tan^2 45^\circ \)
- The tangent of \(45^\circ\) is \(1\).
- Therefore, \( \tan^2 45^\circ = 1^2 = 1\).
- Thus, \(2\tan^2 45^\circ = 2 \times 1 = 2\).
Step 2: Calculate \( \cos^2 30^\circ \)
- The cosine of \(30^\circ\) is \(\frac{\sqrt{3}}{2}\).
- Therefore, \( \cos^2 30^\circ = \left(\frac{\sqrt{3}}{2}\right)^2 = \frac{3}{4}\).
Step 3: Calculate \( \sin^2 60^\circ \)
- The sine of \(60^\circ\) is \(\frac{\sqrt{3}}{2}\).
- Therefore, \( \sin^2 60^\circ = \left(\frac{\sqrt{3}}{2}\right)^2 = \frac{3}{4}\).
Step 4: Combine the Results
Now, substituting the values back into the original expression:
- \( 2\tan^2 45^\circ + \cos^2 30^\circ - \sin^2 60^\circ \)
- \( = 2 + \frac{3}{4} - \frac{3}{4} \)
Final Calculation
- The terms \(\frac{3}{4}\) and \(-\frac{3}{4}\) cancel each other out.
- Thus, the final result is:
\[
2 + 0 = 2
\]
Conclusion
The evaluated expression \(2\tan^2 45^\circ + \cos^2 30^\circ - \sin^2 60^\circ\) equals \(2\).