.

Correct — C. Both fractions are complementary-angle pairs, so each one is 1 before any arithmetic happens.

54° + 36° = 90°, and tan(90° − θ) = cot θ, so tan 54° = cot 36°
→ tan 54°⁄cot 36° = 1

41° + 49° = 90°, and cot(90° − θ) = tan θ, so cot 41° = tan 49°
→ cot 41°⁄tan 49° = 1

Substituting: 2 × 1 − 1 = 1

The options are printed as pictures of numbers, and 1 is option (c).

Why the others are wrong

• (a)
  Option (a) is 0, which is 1 − 1. It drops the coefficient 2 sitting in front of the first fraction and treats the expression as one ratio minus the other.
• (b)
  Option (b) is −1, that is 1 − 2. It attaches the 2 to the second fraction, so the doubled term ends up being subtracted rather than added.
• (d)
  Option (d) is 2. It keeps 2 × 1 and then discards the second fraction, which would only be right if cot 41°⁄tan 49° were 0 instead of 1.

Complementary-angle identities swap a ratio for its co-ratio: tan(90° − θ) = cot θ and cot(90° − θ) = tan θ, with sin/cos and sec/cosec behaving the same way.

So whenever a fraction holds two angles that add to 90°, one on top and its co-ratio below, the fraction is 1 and can be struck out at sight.

The habit worth building is to add the two angles before doing anything else. Here 54 + 36 and 41 + 49 both make 90, and the whole item becomes 2 − 1.

Key facts

• tan(90° − θ) = cot θ, so tan 54° = cot 36° and tan 54°⁄cot 36° = 1.
• cot(90° − θ) = tan θ, so cot 41° = tan 49° and cot 41°⁄tan 49° = 1.
• The pairs to spot are the ones summing to 90°: 54 with 36, and 41 with 49.
• The expression therefore reduces to 2 × 1 − 1 = 1.

Common traps

• Reaching for numeric values of tan 54° instead of checking whether the angles are complements.
• Applying the coefficient 2 to the whole expression rather than to the first fraction only.
• Assuming any two angles adding to 90° cancel, even when the ratio and co-ratio are not paired correctly.

No directly related past PYQ was found.