The value of sin² 30 − sin² 40 + sin² 45 − sin² 55 − sin² 35 + sin² 45 − sin² 50 + sin² 60 is:

Correct — B. The stem is an image: "The value of sin² 30 − sin² 40 + sin² 45 − sin² 55 − sin² 35 + sin² 45 − sin²50 + sin² 60 is:"

Pair each angle with its complement, because sin²x + sin²(90 − x) = 1.

The two negative pairs:
sin²40 + sin²50 = 1 (40 + 50 = 90)
sin²55 + sin²35 = 1 (55 + 35 = 90)
Together they take away 2.

The two positive pairs:
sin²30 + sin²60 = 1/4 + 3/4 = 1
sin²45 + sin²45 = 1/2 + 1/2 = 1
Together they add 2.

2 − 2 = 0 → option (b).

Why the others are wrong

• (a)
  4 — 4 counts all four complementary pairs as additions. Two of them — 40 with 50, and 55 with 35 — sit behind minus signs, so they subtract 2 instead of adding it.
• (c)
  1 — 1 is what is left after cancelling only one of the two negative pairs. There are four minus terms (40, 55, 35, 50) and they make two complete 1s, not one.
• (d)
  2 — 2 is the positive half on its own — sin²30 + sin²60 plus the two sin²45 terms. The four subtracted squares have not been used yet.

Every angle in this line has a partner: 30 with 60, 40 with 50, 55 with 35, and 45 with itself. That is the whole design of the question.

The identity is sin(90 − x) = cos x, so sin²x + sin²(90 − x) = sin²x + cos²x = 1.

Once you see the pairs you never compute a single sine value. You only have to keep the signs straight, because a pair that carries minus signs removes 1 rather than contributing it.

Key facts

• sin(90° − x) = cos x, so sin²x + sin²(90° − x) = 1
• sin 30° = 1/2 and sin 60° = √3/2, so sin²30° + sin²60° = 1/4 + 3/4 = 1
• sin 45° = 1/√2, so sin²45° = 1/2 and two of them make 1
• The complementary pairs here are 30-60, 40-50, 55-35 and 45-45

Common traps

• Pairing the angles correctly but ignoring the minus signs in front of two of the pairs
• Reaching for a calculator value of sin 40° or sin 55° when no numerical value is ever needed
• Missing that sin²45 appears twice and is its own complement

No directly related past PYQ was found.