Question

prove that root 1 minus cos theta ÷ 1+ cos theta = cosec theta minus cot theta​

Answer 1

To Prove

(1 - cos∅)/(1 + cos∅) = (cosec∅ - cot∅)²

Proof

R.H.S → (cosec∅ - cot∅)²

• cosec∅ = 1/sin∅
• cot∅ = cos∅/sin∅

→ (cosec∅ - cot∅)²

→ (1/sin∅ - cos∅/sin∅)²

→ (1 - cos∅)²/sin²∅

→ (1 - cos∅)(1 - cos∅)/(1 - cos²∅)

• 1 - cos²∅ = (1 + cos∅)(1 - cos∅)

→ (1 - cos∅)(1 - cos∅)/(1 + cos∅)(1 - cos∅)

Here, (1 - cos∅) gets cancelled out.

→ (1 - cos∅)/(1 + cos∅) = L.H.S

Q.E.D

Answer 2

Answer:

 R.H.S = ⤵⤵⤵⤵⤵⤵

root 1-cos theta/1+cos theta

= root 1-cos theta/1+cos theta * 1 - cos theta/1-cos theta

= root (1-costheta)^2/ 1- cos^2 theta

= root (1 - cos theta)^2/sin ^2 theta

= 1 - cos theta/sin theta

= 1/sin theta - cos theta/sin theta

= cosec theta - cot theta = L.H.S