Solution:

Given expression: 2 tan^2 45 cos^2 30 - sin^2 60

To simplify the expression, let's break it down step by step.

Step 1: Evaluate the trigonometric functions involved.

tan^2 45 = tan^2 (π/4) = 1

cos^2 30 = cos^2 (π/6) = 3/4

sin^2 60 = sin^2 (π/3) = 3/4

Step 2: Substitute the values obtained in step 1 into the given expression.

2(1)(3/4) - 3/4

Step 3: Simplify the expression further.

2(3/4) - 3/4

= 6/4 - 3/4

= 3/4

Therefore, the simplified form of the expression 2 tan^2 45 cos^2 30 - sin^2 60 is 3/4.

Explanation:

Step 1: Evaluating the trigonometric functions involved
- tan^2 45: The tangent of 45 degrees is equal to 1. Therefore, tan^2 45 = 1.
- cos^2 30: The cosine of 30 degrees is equal to √3/2. Squaring it gives (√3/2)^2 = 3/4.
- sin^2 60: The sine of 60 degrees is equal to √3/2. Squaring it gives (√3/2)^2 = 3/4.

Step 2: Substituting the values obtained in step 1 into the given expression
- 2 tan^2 45 cos^2 30 - sin^2 60 becomes 2(1)(3/4) - 3/4.

Step 3: Simplifying the expression
- Multiplying 2 by 1 gives 2.
- Multiplying 2 by 3/4 gives 6/4.
- Subtracting 3/4 from 6/4 gives 3/4.

Hence, the simplified form of the expression is 3/4.