If sin A - cos A = (√3 - √2) / 2, then find the value of sin A × cos A.

Question

Accepted Answer

Given: sin A - cos A = (√3 - √2) / 2

Step 1: Square both sides:
(sin A - cos A)^2 = ((√3 - √2) / 2)^2

Expanding left side:
sin^2 A - 2 sin A cos A + cos^2 A = (√3 - √2)^2 / 4

Using identity sin^2 A + cos^2 A = 1:
1 - 2 sin A cos A = (3 - 2√6 + 2) / 4

Simplify numerator on right:
1 - 2 sin A cos A = (5 - 2√6) / 4

Step 2: Rearrange to find sin A cos A:
-2 sin A cos A = (5 - 2√6) / 4 - 1
-2 sin A cos A = (5 - 2√6 - 4) / 4
-2 sin A cos A = (1 - 2√6) / 4

Multiply both sides by -1:
2 sin A cos A = (2√6 - 1) / 4

Divide both sides by 2:
sin A cos A = (2√6 - 1) / 8

Therefore, the value of sin A × cos A is (2√6 - 1) / 8.

Answer

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Answer