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Question
Prove the following:
`("cosec"theta + cottheta + 1)/(cottheta + "cosec" theta - 1) = cottheta/("cosec"theta - 1)`
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)`
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