If 1 + cot² θ = cosec² θ, then what is the value of cosec² 60° − cot² 60°?

Correct — A. The stem hands you the identity. Rearrange it, then check it at 60°.

Subtract cot² θ from both sides: cosec² θ − cot² θ = 1
This holds for every θ where cot θ and cosec θ exist, 60° included

sin 60° = √3⁄2, so cosec² 60° = (2⁄√3)² = 4⁄3
cot 60° = 1⁄√3, so cot² 60° = 1⁄3
Subtract: 4⁄3 − 1⁄3 = 3⁄3 = 1 → option (a)

Why the others are wrong

• (b)
  2 — The difference cannot be 2. cosec² 60° = 4⁄3 and cot² 60° = 1⁄3 differ by exactly 1. Getting 2 would need cosec² 60° = 7⁄3.
• (c)
  3 — 3 is tan² 60°, a single squared ratio, not a difference. cosec² θ and cot² θ always sit exactly 1 apart, at 60° as at any angle where both exist.
• (d)
  4 — 4 is cosec² 30°, since cosec 30° = 2. Even at 30° the difference is 1: cosec² 30° − cot² 30° = 4 − 3. Changing the angle changes the terms, never the difference.

All three Pythagorean identities come from sin² θ + cos² θ = 1.

Divide it by sin² θ: 1 + cot² θ = cosec² θ
Divide it by cos² θ: tan² θ + 1 = sec² θ

Rearranged, each gives a difference that is always 1: cosec² θ − cot² θ = 1 and sec² θ − tan² θ = 1. The angle does not matter, provided the ratios are defined.

Key facts

• cosec² θ − cot² θ = 1 wherever sin θ ≠ 0.
• sec² θ − tan² θ = 1 wherever cos θ ≠ 0.
• sin 60° = √3⁄2, so cosec 60° = 2⁄√3 and cosec² 60° = 4⁄3.
• cot 60° = 1⁄√3, so cot² 60° = 1⁄3.

Common traps

• Squaring 2⁄√3 as 2⁄3. Both parts square: (2⁄√3)² = 4⁄3.
• Pairing the wrong ratios. cosec goes with cot and sec goes with tan: cosec² 60° − tan² 60° = 4⁄3 − 3, not 1.

No directly related past PYQ was found.