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If θtan θ=2021 show that 1-sinθ+cosθ1+sinθ+cosθ=37 - Mathematics

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प्रश्न

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

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उत्तर

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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पाठ 10: Trigonometric Ratios - Exercise 10.1 [पृष्ठ २५]

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आरडी शर्मा Mathematics [English] Class 10
पाठ 10 Trigonometric Ratios
Exercise 10.1 | Q 31 | पृष्ठ २५

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