Find the value of (sec 35°) ⁄ (cosec 55°) − (cos 25°) ⁄ (sin 65°).

Correct — B. Option (b) shows 0.

The stem is printed as an image and reads: find the value of sec 35° ⁄ cosec 55° − cos 25° ⁄ sin 65°.

Rule: cosec (90° − θ) = sec θ and sin (90° − θ) = cos θ.

cosec 55° = cosec (90° − 35°) = sec 35°, so the first fraction is sec 35° ⁄ sec 35° = 1
sin 65° = sin (90° − 25°) = cos 25°, so the second fraction is cos 25° ⁄ cos 25° = 1

1 − 1 = 0 → option (b)

Why the others are wrong

• (a)
  Option (a) shows 2, the value you get by adding the two fractions. The sign printed between them in the stem is a minus.
• (c)
  Option (c) shows 1, which is each fraction taken on its own. Stopping after the first ratio leaves the second half of the expression unused.
• (d)
  Option (d) shows 1⁄2. Nothing here can produce a half: each fraction has the same ratio of the same angle on top and bottom, so both are exactly 1.

Complementary-angle identities convert any ratio of θ into the co-ratio of 90° − θ: sin ↔ cos, tan ↔ cot, sec ↔ cosec.

The examiner's whole design here is the pairing. 35° + 55° = 90° and 25° + 65° = 90°, so each fraction is a ratio divided by itself.

Nothing needs to be evaluated. Scan a trigonometric expression for angle pairs summing to 90° before you reach for values, because that is usually where the whole sum collapses.

Key facts

• sin (90° − θ) = cos θ and cos (90° − θ) = sin θ.
• sec (90° − θ) = cosec θ, so cosec 55° is the same number as sec 35°.
• 35° + 55° = 90° and 25° + 65° = 90°, which is why both fractions equal 1.

Common traps

• Reaching for decimal values of sec 35° and cosec 55° instead of pairing the angles
• Reading the minus between the two fractions as a plus and answering 2

No directly related past PYQ was found.