Question

Que :-

(cos ∅ × cosec theta - sin ∅ × sec theta) / (cos ∅ + sin ∅) = cosec ∅ - sec ∅​

Answer 1

Answer:

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Step-by-step explanation:

$to \: prove : \\ \frac{cosθ \times cosecθ \: - sin θ \times secθ}{cosθ + sinθ} = cosecθ - secθ \\ \\ LHS = \frac{cosθ \times cosecθ - sinθ \times secθ}{cosθ + sinθ} \\ \\ = \frac{cosθ \times \frac{1}{sinθ} - sinθ \times \frac{1}{cosθ} }{cosθ + sinθ} \\ \\ = \frac{ \frac{cosθ}{sinθ} - \frac{sinθ}{cosθ} }{cosθ + sinθ} \\ \\ = \frac{cos {}^{2} θ - sin {}^{2}θ }{sinθ.cosθ(cosθ + sinθ)} \\ \\ = \frac{(cosθ + sinθ)(cosθ - sinθ)}{sinθ.cosθ(cosθ + sinθ)} \\ \\ = \frac{cosθ - sinθ}{sinθ.cosθ} \\ \\ = \frac{cosθ}{sinθ.cosθ} - \frac{sinθ}{sinθ.cosθ} \\ \\ = \frac{1}{sinθ} - \frac{1}{cosθ} \\ \\ = cosecθ - secθ \\ \\ = RHS$