Question

Cos theta minus sin theta plus one by cos theta plus sin theta minus one . Prove that is equal to cosectheta plus cot theta

Answer 1

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Answer 2

(Cosθ - Sinθ + 1)/(Cosθ + Sinθ - 1)  = Cosecθ + Cotθ

Step-by-step explanation:

(Cosθ - Sinθ + 1)/(Cosθ + Sinθ - 1)  = Cosecθ + Cotθ

LHS

= (Cosθ - Sinθ + 1)/(Cosθ + Sinθ - 1)

=  (Cosθ + (1 - Sinθ))/(Cosθ - (1 - Sinθ))

Multiplying & Dividing by  (Cosθ + (1 - Sinθ)

= (Cos²θ + 1 + Sin²θ -2Sinθ + 2Cosθ(1 - Sinθ))/( Cos²θ - (1 + Sin²θ - 2Sinθ))

using Cos²θ + Sin²θ = 1

= (2  -2Sinθ + 2Cosθ(1 - Sinθ))/( ( Cos² - (Cos²θ + Sin²θ + Sin²θ - 2Sinθ))

= (2(1 - Sinθ) + 2Cosθ(1 - Sinθ))/(2Sinθ - 2Sin²θ)

= (2 + 2Cosθ)(1 - Sinθ)/(2Sinθ(1 - Sinθ))

= 2(1 + Cosθ)(1 - Sinθ)/(2Sinθ(1 - Sinθ))

= (1 + Cosθ)/Sinθ

= 1/Sinθ + Cosθ/Sinθ

= Cosecθ + Cotθ

= RHS

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