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Prove the following: cosecθ+cotθ+1cotθ+cosecθ-1=cotθcosecθ-1 - Mathematics and Statistics

Sum

Prove the following:

`("cosec"theta + cottheta + 1)/(cottheta + "cosec" theta - 1) = cottheta/("cosec"theta - 1)`

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Solution

We know that,

1 + cot2θ = cosec2θ

∴ cot2θ = cosec2θ – 1

∴ cotθ·cotθ = (cosecθ – 1)( cosecθ + 1)

∴ `cottheta/("cosec"theta - 1) = ("cosec"theta + 1)/cottheta`

By the theorem on equal ratios, we get

∴ `(cot theta)/("cosec"theta-1)=("cosec"theta + 1)/(cottheta) = (cot theta + "cosec"theta+1)/("cosec" theta-1+cottheta)`

∴ `("cosec"theta + cottheta + 1)/(cottheta + "cosec" theta - 1) = cottheta/("cosec"theta - 1)`

Concept: Fundamental Identities
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APPEARS IN

Balbharati Mathematics and Statistics 1 (Arts and Science) 11th Standard Maharashtra State Board
Chapter 2 Trigonometry - 1
Miscellaneous Exercise 2 | Q 10. (xviii) | Page 34
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