If sin x = cos x, find sin⁴ x + cos⁴ x.

Correct — C.
Given sin x = cos x, square both: sin²x = cos²x
Use sin²x + cos²x = 1: 2sin²x = 1, so sin²x = cos²x = 1⁄2

Square again: sin⁴x = cos⁴x = (1⁄2)² = 1⁄4
Add: 1⁄4 + 1⁄4 = 1⁄2
Check with x = 45°: sin45° = cos45° = 1⁄√2, and (1⁄√2)⁴ = 1⁄4
So sin⁴x + cos⁴x = 1⁄2 → option (c)

Why the others are wrong

• (a)
  1 — 1 is sin²x + cos²x, not the fourth powers. sin⁴x + cos⁴x reaches 1 only when sin x or cos x is 0, which cannot happen when the two are equal.
• (b)
  3⁄4 — 3⁄4 would need sin²x × cos²x = 1⁄8, since sin⁴x + cos⁴x = 1 − 2sin²xcos²x. Here sin²x × cos²x = 1⁄2 × 1⁄2 = 1⁄4, which gives 1⁄2.
• (d)
  1⁄4 — 1⁄4 is sin⁴x alone, or cos⁴x alone. The question adds the two equal fourth powers: 1⁄4 + 1⁄4 = 1⁄2.

When sin x = cos x, tan x = 1, and the two squares split their total of 1 equally: sin²x = cos²x = 1⁄2. For an acute angle that is x = 45°.

Fourth powers are best reached through squares: sin⁴x + cos⁴x = (sin²x + cos²x)² − 2sin²xcos²x = 1 − 2sin²xcos²x.

That expression is smallest, 1⁄2, exactly when sin²x = cos²x, and largest, 1, when sin x or cos x is 0.

Key facts

• sin x = cos x gives tan x = 1, so x = 45° for an acute angle
• sin45° = cos45° = 1⁄√2
• sin⁴x + cos⁴x = 1 − 2sin²xcos²x
• sin⁴x + cos⁴x always lies between 1⁄2 and 1, inclusive

Common traps

• Stopping at sin²x + cos²x = 1 and answering 1
• Taking sin45° as 1⁄2 (that is sin30°), which gives 1⁄16 + 1⁄16 = 1⁄8

No directly related past PYQ was found.