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Question
Prove the following:
`(1 + cottheta + "cosec" theta)/(1 - cottheta + "cosec" theta) = ("cosec" theta + cottheta - 1)/(cottheta - "cosec"theta + 1)`
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Solution
We know that, cosec2θ – cot2θ = 1
∴ (cosecθ – cotθ)(cosecθ + cotθ) = 1
∴ `("cosec" theta + cottheta)/1 = 1/("cosec" theta - cottheta)`
By componendo-dividendo, we get
`("cosec" theta + cottheta+1)/("cosec" theta + cottheta-1) = (1+"cosec"theta-cottheta)/(1-("cosec" theta - cottheta))`
∴ `("cosec" theta + cottheta + 1)/("cosec" theta + cottheta -1)=(1 + "cosec"theta - cottheta)/(1 - "cosec"theta + cottheta)`
∴ `("cosec" theta+cot theta + 1)/(1 +"cosec" theta-cottheta) = ("cosec" theta + cottheta - 1)/(cottheta - "cosec"theta + 1)`
LHS = RHS
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