Question

cos theta - sin theta +1/ sin theta +cos theta -1 =1/cosec theta - cot theta. Step by step explaination no short cut​

Answer 1

Answer:

Step-by-step explanation:

Cosθ - Sinθ + 1 / Sinθ+Cosθ - 1

//Multiply and divide by Sinθ+Cosθ+1

=> [Cosθ - Sinθ + 1 / Sinθ+Cosθ - 1]  * [Sinθ+Cosθ+1/Sinθ+Cosθ + 1]

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

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

=> 1 + Cos²θ + 2Cosθ - Sin²θ / Sin²θ + Cos²θ + 2SinθCosθ - 1

=> 2Cos²θ + 2Cosθ / 2SinθCosθ   (∵ Sin²θ + Cos²θ = 1)

=> 2Cosθ(Cosθ + 1) / 2SinθCosθ

=> Cosθ + 1 / Sinθ

= Cosecθ + Cotθ

//Multiply and divide by Cosecθ - Cotθ

= Cosecθ + Cotθ  * [Cosecθ - Cotθ / Cosecθ - Cotθ]

= Cosec²θ - Cot²θ / Cosecθ - Cotθ

= 1/Cosecθ - Cotθ    ( ∵Cosec²θ - Cot²θ = 1)

= R.H.S

Hence Proved.