If then the value of is:

Correct — A. 59° and 31° add to 90°, so convert the cotangent before doing anything else.

cot 59° = cot(90° − 31°) = tan 31°

sin 31° = α, and 31° is acute, so
cos 31° = √(1 − α²)

tan 31° = sin 31° ⁄ cos 31°
= α ⁄ √(1 − α²) → option (a).

Why the others are wrong

• (b)
  α ⁄ √(1 + α²) carries a plus sign under the root. That form appears when α is a tangent; here α is a sine, so the cosine is √(1 − α²).
• (c)
  √(1 + α²) ⁄ α is option (b) turned upside down. It keeps the wrong sign under the root and inverts the ratio.
• (d)
  √(1 − α²) ⁄ (2α) flips the ratio, which gives cot 31° (that is, tan 59°) rather than tan 31°. The 2 in the denominator answers to nothing in the question.

This is a co-function item. The complementary-angle rules are sin θ = cos(90° − θ), tan θ = cot(90° − θ) and sec θ = cosec(90° − θ).

Because 59° + 31° = 90°, cot 59° is exactly tan 31°, and the question collapses to writing tan 31° in terms of sin 31°.

The second tool is the Pythagorean identity. From sin²θ + cos²θ = 1, cos 31° = √(1 − α²), taken positive because 31° lies in the first quadrant.

Key facts

• cot(90° − θ) = tan θ, so cot 59° = tan 31°.
• If sin θ = α and θ is acute, then cos θ = √(1 − α²) and tan θ = α ⁄ √(1 − α²).
• Two angles are complementary when they sum to 90°, and 59° + 31° = 90°.

Common traps

• Turning cot 59° into tan 59° by dropping the co- prefix without the 90° − θ step
• Writing cos 31° as √(1 + α²) instead of √(1 − α²)

No directly related past PYQ was found.