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Prove: `(tan theta)/(sec theta-1)=(tan theta+sec theta+1)/(tan theta+sec theta-1)`

Answer» Proof: `1+tan^(2)theta=sec^(2)theta` …(Identity)
`:.tan^(2) theta=sec^(2)theta-1`
`:.tan theta xx ta theta=(sec theta+1)(sec theta-1)`
`:.(tan theta)/((sec theta-1))=((sec theta+1))/(tan theta)`
By theorem on equal ratios,
`(tan theta)/((sec theta-1))=((sec theta+1))/(tan theta)=(tan theta+sec theta+1)/(sec theta-1+tan theta)`
`:.(tan theta)/(sec theta-1)=(tan theta+sec theta+1)/(tan theta+sec theta-1)`


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