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प्रश्न
Prove the following:
`("cosec"theta + cottheta - 1)/( "cosec"theta + cot theta + 1) =(1-sintheta)/costheta`
उत्तर
We know that,
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
`cottheta/("cosec"theta + 1) = ("cosec"theta - 1) /cottheta = (cottheta + "cosec"theta - 1)/("cosec"theta +1 + cottheta)`
∴ `("cosec"theta- 1)/(cottheta) = (cottheta+"cosec"theta-1)/("cosec"theta + 1 + cot theta)`
∴ `(1/sintheta-1)/(costheta/sintheta)=("cosec"theta+cottheta-1)/("cosec"theta+cottheta+1)`
∴ `("cosec"theta + cottheta - 1)/( "cosec"theta + cot theta + 1) =(1-sintheta)/costheta`
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