Which of the following is true for all acute angles A?

Correct — B.
In a right triangle the two acute angles are A and 90° − A.
The side opposite A is the side adjacent to 90° − A.

sin A = opposite⁄hypotenuse
= adjacent (to 90° − A)⁄hypotenuse
= cos(90° − A) → option (b).

Check at A = 30°: sin 30° = 1⁄2 = cos 60°.

Why the others are wrong

• (a)
  sin A = sin(90° − A) — sin(90° − A) is cos A, not sin A. The two agree only at A = 45°. At 30°, sin 30° = 1⁄2 but sin 60° = √3⁄2.
• (c)
  cos A = tan(90° − A) — tan(90° − A) is cot A, not cos A. At A = 45°, cos 45° = 1⁄√2 while tan 45° = 1.
• (d)
  cot A = sin(90° − A) — sin(90° − A) is cos A, not cot A. At A = 60°, cot 60° = 1⁄√3 while sin 30° = 1⁄2.

Replacing A by 90° − A swaps each function for its co-function: sine with cosine, tangent with cotangent, secant with cosecant. The reason is the right triangle: the side opposite one acute angle is adjacent to the other.

So the swap changes the function's name as well as the angle. An identity that keeps the name (sin with sin) or pairs the wrong partners (cos with tan) cannot hold for every acute A.

Key facts

• sin(90° − A) = cos A and cos(90° − A) = sin A.
• tan(90° − A) = cot A and cot(90° − A) = tan A.
• sec(90° − A) = cosec A and cosec(90° − A) = sec A.
• Among acute angles, sin A = cos A only at A = 45°.

Common traps

• Keeping the function name when the angle becomes 90° − A, as option (a) does.
• Pairing tangent with cosine, as option (c) does. Tangent's co-function is cotangent.

No directly related past PYQ was found.