Find the value of (cosec θ - sin θ)(sec θ - cos θ)(tan θ + cot θ).

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Step 1: Write the expression:
(cosec θ - sin θ)(sec θ - cos θ)(tan θ + cot θ)

Step 2: Use trigonometric identities:
cosec θ = 1/sin θ
sec θ = 1/cos θ

Step 3: Simplify each bracket:
(cosec θ - sin θ) = (1/sin θ - sin θ) = (1 - sin^2 θ)/sin θ = cos^2 θ / sin θ
(sec θ - cos θ) = (1/cos θ - cos θ) = (1 - cos^2 θ)/cos θ = sin^2 θ / cos θ

Step 4: Simplify (tan θ + cot θ):
tan θ = sin θ / cos θ
cot θ = cos θ / sin θ
So, tan θ + cot θ = (sin θ / cos θ) + (cos θ / sin θ) = (sin^2 θ + cos^2 θ) / (sin θ cos θ) = 1 / (sin θ cos θ)

Step 5: Multiply all three:
= (cos^2 θ / sin θ) * (sin^2 θ / cos θ) * (1 / (sin θ cos θ))
= (cos^2 θ * sin^2 θ) / (sin θ * cos θ) * (1 / (sin θ cos θ))
= (cos^2 θ * sin^2 θ) / (sin θ * cos θ) * (1 / (sin θ cos θ))

Simplify numerator and denominator:
= (cos^2 θ * sin^2 θ) / (sin θ * cos θ * sin θ * cos θ)
= (cos^2 θ * sin^2 θ) / (sin^2 θ * cos^2 θ) = 1

Therefore, the value of the expression is 1.

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