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