(cot θ + tan θ)(cosec θ − sin θ)(cos θ − sec

Solution

We start by simplifying each term individually: ⇒ cot θ + tan θ = (cos θ/sin θ) + (sin θ/cos θ) ⇒ = (cos² θ + sin² θ) / (sin θ cos θ) ⇒ = 1 / (sin θ cos θ)            [since cos² θ + sin² θ = 1] Next, simplify cosec θ - sin θ: ⇒ cosec θ - sin θ = (1/sin θ) - sin θ ⇒ = (1 - sin² θ) / sin θ ⇒ = cos² θ / sin θ            [since 1 - sin² θ = cos² θ] Finally, simplify cos θ - sec θ: ⇒ cos θ - sec θ = cos θ - (1/cos θ) ⇒ = (cos² θ - 1) / cos θ ⇒ = -sin² θ / cos θ            [since cos² θ - 1 = -sin² θ] Now, combine all three simplified terms: ⇒ (cot θ + tan θ)(cosec θ - sin θ)(cos θ - sec θ) ⇒ = (1 / (sin θ cos θ)) × (cos² θ / sin θ) × (-sin² θ / cos θ) ⇒ = (1 / (sin θ cos θ)) × (cos² θ / sin θ) × (-sin² θ / cos θ) ⇒ = (-1)    [After simplifying the terms and canceling out common factors] ∴ The expression equals -1.