Question

If `tan θ = 20/21` show that `(1 - sin theta + cos theta)/(1 + sin theta + cos theta) = 3/7`

Answer 1

Given: tan θ = `20/21`,

show that `(1 - sin theta + cos theta)/(1 + sin theta + cos theta) = 3/7`

Since tan θ = perpendicular/base

So we construct right triangle ABC right angled at C

such that ∠ABC = θ and AC = Perpendicular = 20

BC = base = 21

By Pythagoras theorem, AB² = AC² + BC²

⇒ AB² = (20)² + (21)²

⇒ AB² = 400 + 441

⇒ AB² = 841

⇒ AB = `sqrt841`

⇒ AB = 29

As sin θ = perpendicular / hypotenuse cos θ = base / hypotenuse

So, 

tan θ = `20/21` ⇒ sin θ `20/29 and cos θ = 21/29`

∴ `(1 - sin θ + cos θ)/(1 + sin θ + cos θ) = (1 - 20/29 + 21/29)/(1 + 20/29 + 21/29)`

= `(30/29)/(70/29)`

= `3/7`  Hence proved