If sin A − cos A = (√3 − √2)⁄2, then the value of sin A. cos A is:

Correct — B. The stem gives sin A − cos A = (√3 − √2) ⁄ 2 and wants sin A · cos A.

Rule: square the given difference, because sin²A + cos²A = 1 makes the squares collapse.

(sin A − cos A)² = sin²A + cos²A − 2 sin A cos A = 1 − 2 sin A cos A
((√3 − √2) ⁄ 2)² = (3 − 2√6 + 2) ⁄ 4 = (5 − 2√6) ⁄ 4

Set the two equal:
2 sin A cos A = 1 − (5 − 2√6) ⁄ 4 = (2√6 − 1) ⁄ 4

sin A cos A = (2√6 − 1) ⁄ 8 → option (b), the image reading (2√6 − 1) ⁄ 8

Why the others are wrong

• (a)
  Option (a) shows (2√2 − 1) ⁄ 8. The denominator is right, the surd is not — the cross term of (√3 − √2)² is 2 × √3 × √2 = 2√6, not 2√2.
• (c)
  Option (c) shows (3√2 − 1) ⁄ 4. Neither part survives a check: the surd should be √6, and the final denominator is 8, since you divide by 4 when squaring and then halve to isolate the product.
• (d)
  Option (d) shows (2√3 − 1) ⁄ 4. It keeps √3 from the numerator instead of forming √6, and it stops at the 2 sin A cos A stage, leaving 4 in the denominator where 8 belongs.

Every 'given sin ± cos, find sin · cos' item is one identity wearing a disguise.

Squaring either combination turns the pair into 1 ± 2 sin A cos A, because sin²A + cos²A = 1 absorbs both squares. The product you are asked for is then half the difference between 1 and the square of what you were given.

You never need the angle A, and you should not go looking for it. Here (√3 − √2) ⁄ 2 is about 0.159 and belongs to no standard angle.

Key facts

• (sin A − cos A)² = 1 − 2 sin A cos A.
• (sin A + cos A)² = 1 + 2 sin A cos A.
• (√3 − √2)² = 5 − 2√6, because the cross term is 2 × √3 × √2 = 2√6.
• The answer (2√6 − 1) ⁄ 8 is about 0.487, a legal value since sin A cos A cannot exceed 0.5.

Common traps

• Expanding (√3 − √2)² as 3 − 2 and losing the middle term entirely
• Stopping at 2 sin A cos A and reading that off as the answer
• Hunting for the angle A instead of treating the product as the unknown

No directly related past PYQ was found.