If cotθ = 4⁄3, then evaluate (secθ(1 + cot²θ)(cosec²θ − cot²θ)) ⁄ cosec³θ.

Correct — C. The question runs across two images: the first gives cot θ = 4⁄3, the second the expression (sec θ (1 + cot²θ)(cosec²θ − cot²θ)) ⁄ cosec³θ.

Two identities empty most of it out:
1 + cot²θ = cosec²θ
cosec²θ − cot²θ = 1

Substitute both:
(sec θ × cosec²θ × 1) ⁄ cosec³θ
= sec θ ⁄ cosec θ = tan θ

cot θ = 4⁄3, so tan θ = 3⁄4 → option (c), the picture reading 3/4.

Why the others are wrong

• (a)
  4⁄5 is cos θ for this triangle — adjacent 4, hypotenuse 5. The expression reduces to sec θ ⁄ cosec θ, which is tan θ, so cos θ stays in the denominator rather than becoming the answer.
• (b)
  3⁄5 is sin θ for this triangle. sec θ ⁄ cosec θ = (1 ⁄ cos θ) × sin θ = sin θ ⁄ cos θ, so the hypotenuse cancels out and 5 cannot appear in the answer.
• (d)
  4⁄3 is cot θ itself, the value the stem gives you. The expression collapses to tan θ, its reciprocal, so 4⁄3 answers a question that was not asked.

Two Pythagorean identities do all the work.

1 + cot²θ = cosec²θ turns the first bracket into cosec²θ, and cosec²θ − cot²θ = 1 makes the second bracket disappear.

What is left is sec θ × cosec²θ ⁄ cosec³θ = sec θ ⁄ cosec θ, and that ratio is tan θ.

The lesson is order. Simplify the identities first and substitute the number last; substituting cot θ = 4⁄3 at the start turns a two-line problem into a page of fractions that reaches the same 3⁄4.

Key facts

• 1 + cot²θ = cosec²θ, the Pythagorean identity divided through by sin²θ.
• cosec²θ − cot²θ = 1 wherever both ratios are defined.
• sec θ ⁄ cosec θ = tan θ, and cosec θ ⁄ sec θ = cot θ.
• cot θ = 4⁄3 corresponds to a right triangle with legs 4 and 3 and hypotenuse 5.

Common traps

• Substituting 4⁄3 before simplifying, which buries the cancellation in fraction arithmetic.
• Reporting cot θ = 4⁄3 after correctly reducing the expression to tan θ.
• Solving only the first image and never noticing the sentence continues.

No directly related past PYQ was found.