What is the value of 3⁄2 ((cos 39)⁄(sin 51)) − √(sin² 39° + sin² 51°) ?

Correct — D. 39° and 51° add to 90°, so each is the complement of the other.

Complement rule: cos 39° = sin (90° − 39°) = sin 51°
First term: 3⁄2 × (sin 51° ⁄ sin 51°) = 3⁄2 × 1 = 3⁄2

Complement rule again: sin 51° = cos 39°
Under the root: sin² 39° + cos² 39° = 1
Root: √1 = 1

Subtract: 3⁄2 − 1 = 1⁄2 → option (d)

Why the others are wrong

• (a)
  3⁄2 — 3⁄2 is the first term alone. The root is not zero: sin² 39° + sin² 51° = sin² 39° + cos² 39° = 1, so 1 must still be subtracted.
• (b)
  7⁄8 — 7⁄8 would need the root to equal 5⁄8. Both parts are exact — the ratio is 1 and the root is √1 = 1 — so the value is 3⁄2 − 1.
• (c)
  5⁄6 — 5⁄6 would need the root to equal 2⁄3, so the sum under it would be 4⁄9. But sin² 39° + sin² 51° is sin² 39° + cos² 39°, exactly 1.

Two facts carry this item. Complementary angles swap the ratios: sin (90° − θ) = cos θ, and in the same way tan and cot, sec and cosec swap.

The Pythagorean identity, sin² θ + cos² θ = 1, then clears a sum of the squares of a sine and a cosine of the same angle.

When the angles in an expression pair up to 90°, rewrite one angle of each pair first. Here that turns both parts into the number 1.

Key facts

• sin (90° − θ) = cos θ and cos (90° − θ) = sin θ.
• sin² θ + cos² θ = 1 for every angle θ.
• cos 39° ⁄ sin 51° = 1, because 39° + 51° = 90°.

Common traps

• Reaching for approximate values of sin 51° and cos 39° instead of spotting that they are equal.
• Stopping at 3⁄2 once the fraction becomes 1, and dropping the root term.

No directly related past PYQ was found.