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Question

Accepted Answer

Given: cotθ = 4/3.

Let's use a right triangle definition:
Let the adjacent side = 4k, opposite side = 3k (since cotθ = adjacent/opposite).
Hypotenuse = √(4k)^2 + (3k)^2 = √(16k^2 + 9k^2) = √25k^2 = 5k.

Now, compute the required trigonometric values:
- sinθ = opposite/hypotenuse = 3k/5k = 3/5
- cosθ = adjacent/hypotenuse = 4/5
- secθ = 1/cosθ = 5/4
- cotθ = 4/3
- cosecθ = 1/sinθ = 5/3

Now, plug into the expression:
(secθ)(1 + cot²θ)(cosec²θ − cot²θ) / (cosec³θ)

First, compute each part:
1. cot²θ = (4/3)^2 = 16/9
2. 1 + cot²θ = 1 + 16/9 = 25/9
3. cosec²θ = (5/3)^2 = 25/9
4. cosec²θ − cot²θ = 25/9 − 16/9 = 9/9 = 1
5. secθ = 5/4
6. cosec³θ = (5/3)^3 = 125/27

Now substitute:
Numerator = secθ × (1 + cot²θ) × (cosec²θ − cot²θ) = (5/4) × (25/9) × 1 = (5 × 25)/(4 × 9) = 125/36
Denominator = cosec³θ = 125/27

So, the whole expression = (125/36) ÷ (125/27) = (125/36) × (27/125) = 27/36 = 3/4

Final Answer: 3/4

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