IfsinA = x, then what is cos²A in terms of x?

Correct — A. Use the identity that links sin and cos of the same angle.

Pythagorean identity: sin²A + cos²A = 1
Subtract sin²A: cos²A = 1 − sin²A
Substitute sin A = x: cos²A = 1 − x²

Check with A = 30°: x = 1⁄2, 1 − x² = 3⁄4, and cos²30° = (√3⁄2)² = 3⁄4 ✓

cos²A = 1 − x² → option (a)

Why the others are wrong

• (b)
  2 − x² — 2 − x² would make sin²A + cos²A equal 2. At A = 0°, x = 0 and it gives cos²A = 2, but cos²A never exceeds 1.
• (c)
  3 − x² — 3 − x² fails the same test: at x = 0 it gives cos²A = 3, while cos²0° = 1.
• (d)
  1 − 2x² — 1 − 2x² is cos 2A (the double-angle formula 1 − 2sin²A), not cos²A. At A = 30° it gives 1⁄2, while cos²30° = 3⁄4.

For every angle A, sin²A + cos²A = 1. In a right triangle with hypotenuse 1, the two legs are sin A and cos A, and Pythagoras gives the identity.

So knowing one ratio fixes the square of the other. cos A itself would be ±√(1 − x²), the sign set by the quadrant A lies in; cos²A carries no such ambiguity.

Key facts

• sin²A + cos²A = 1 for every angle A.
• cos 2A = 1 − 2sin²A = 2cos²A − 1, which is not the same as cos²A.
• The other two Pythagorean identities are 1 + tan²A = sec²A and 1 + cot²A = cosec²A.

Common traps

• Taking 1 − 2x², which is cos 2A, for cos²A
• Answering √(1 − x²), which is cos A up to sign, when the question asks for cos²A

No directly related past PYQ was found.