If cosec θ = 5⁄3, then evaluate (sec²θ − 1) × cot²θ × (1 + cot²θ).

Correct — B. The stem and all four options are images. The stem reads: If cosecθ = 5/3, then evaluate (sec²θ − 1) × cot²θ × (1 + cot²θ). Option (b) shows 25/9.

cosecθ = 5/3 means sinθ = 3/5, so the triangle is 3-4-5 and cosθ = 4/5.
That gives tanθ = 3/4 and cotθ = 4/3.

Take the brackets one at a time:
sec²θ − 1 = tan²θ = 9/16
cot²θ = 16/9
1 + cot²θ = cosec²θ = 25/9

9/16 × 16/9 = 1, so the product collapses to 25/9 → option (b).

Why the others are wrong

• (a)
  9/16 is only the first bracket, sec²θ − 1 = tan²θ. Two brackets are still unmultiplied, and together they contribute 16/9 × 25/9.
• (c)
  25/16 is sec²θ, which is 1 + tan²θ. The third bracket is 1 + cot²θ, and that identity gives cosec²θ = 25/9, not sec²θ.
• (d)
  4/5 is cosθ, read straight off the 3-4-5 triangle. It is a value from the setup, not the value of the product the question asks for.

Three Pythagorean identities are in play, and two of them cancel each other.

sec²θ − 1 = tan²θ, and tan²θ × cot²θ = 1 for every θ, so the first two brackets multiply to 1 no matter what cosecθ is.

That leaves 1 + cot²θ, which is cosec²θ by the third identity. Since cosecθ = 5/3 is given, the answer is (5/3)² = 25/9 without ever finding θ.

Seeing that shortcut turns a three-bracket grind into one squaring.

Key facts

• 1 + tan²θ = sec²θ, so sec²θ − 1 = tan²θ
• 1 + cot²θ = cosec²θ
• tanθ × cotθ = 1, so tan²θ × cot²θ = 1
• cosecθ = 5/3 gives sinθ = 3/5, cosθ = 4/5, tanθ = 3/4 and cotθ = 4/3

Common traps

• Stopping after the first bracket and answering tan²θ
• Using 1 + tan²θ where the expression writes 1 + cot²θ
• Computing all three brackets in decimals and losing the exact fraction

No directly related past PYQ was found.