The greatest value of sin⁴θ + cos⁴θ is:

Correct — A. The stem asks for the greatest value of sin⁴θ + cos⁴θ. Turn the fourth powers into second powers first.

sin⁴θ + cos⁴θ = (sin²θ + cos²θ)² − 2sin²θcos²θ
= 1 − 2sin²θcos²θ

Since 2 sin θ cos θ = sin 2θ, we have 2sin²θcos²θ = ½ sin²2θ.
Expression = 1 − ½ sin²2θ

To make that as large as possible, make sin²2θ as small as possible, and its least value is 0.
Greatest value = 1 − 0 = 1 → option (a).

Why the others are wrong

• (b)
  3 — 3 is outside the range entirely. sin⁴θ and cos⁴θ each lie between 0 and 1, so their sum cannot pass 2, and the identity caps it at 1.
• (c)
  4 — 4 would need each term to reach 2, but sin²θ and cos²θ never exceed 1 and neither do their squares. The expression in fact never rises above 1.
• (d)
  2 — 2 is what you get by letting sin⁴θ and cos⁴θ both hit 1 at the same angle. They cannot: sin²θ + cos²θ = 1 forces one to shrink as the other grows.

One identity does the work: sin²θ + cos²θ = 1. Writing a = sin θ and b = cos θ, a⁴ + b⁴ = (a² + b²)² − 2a²b², and the bracket is 1.

That leaves 1 − 2sin²θcos²θ, so the whole range question reduces to a question about the single quantity sin²θcos²θ.

Because 2 sin θ cos θ = sin 2θ, that quantity equals ¼ sin²2θ, which runs from 0 to ¼. So sin⁴θ + cos⁴θ runs from ½ to 1, and 1 is the greatest value.

Key facts

• sin⁴θ + cos⁴θ = 1 − 2sin²θcos²θ for every θ.
• The greatest value of sin⁴θ + cos⁴θ is 1, taken at θ = 0°, 90°, 180° and 270°.
• Its least value is ½, taken at 45° and at every odd multiple of 45°.

Common traps

• Assuming sin θ and cos θ can both equal 1, which would give the sum 2.
• Reporting ½, the least value, when the stem asks for the greatest.
• Squaring sin²θ + cos²θ = 1 and forgetting the −2sin²θcos²θ cross term.

No directly related past PYQ was found.