If \( \sin(x) = a \) and \( \cos(x) = b \), express the expression \( 2\sin(x)\cos(x) - 2a^2 + 2b^2 \) in its simplest form.
Answer
Step 4: Distribute the 2 in the second part of the expression to get \( \sin(2x) + 2 - 4a^2 \)
Step 1 ：Step 1: Use the trigonometric identity \( \sin(2x) = 2\sin(x)\cos(x) \) to simplify the first part of the expression. So the expression becomes \( \sin(2x) - 2a^2 + 2b^2 \)
Step 2 ：Step 2: Group the terms to factor by grouping. The expression now becomes \( \sin(2x) + 2(b^2 - a^2) \)
Step 3 ：Step 3: Use the Pythagorean trigonometric identity \( \sin^2(x) + \cos^2(x) = 1 \) to replace \( b^2 - a^2 \) with \( 1 - 2a^2 \). The expression now simplifies to \( \sin(2x) + 2(1 - 2a^2) \)
Step 4 ：Step 4: Distribute the 2 in the second part of the expression to get \( \sin(2x) + 2 - 4a^2 \)
