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l tan theta + m sec theta = n ,l' tan t...

`l tan theta + m sec theta = n` ,`l' tan theta - m' sec theta = n' ` Prove `[(nl' - ln')/( ml' + m'l)]^2 = 1 + [(m'n+mn')/(lm' + l'm)]^2`

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`l tantheta + m sectheta = n->(1)`
`l' tantheta - m' sectheta = n'->(2)`
Multiplying (1) with `m'` and (2) with `m` and then adding both,
`lm' tan theta +mm'sec theta + ml' tantheta - mm' sectheta = nm'+mn'`
`=>lm' tan theta + ml' tantheta = nm'+mn'`
`=>tan theta = ( nm'+mn')/(lm' + ml')`
Multiplying (1) with `l'` and (2) with `l` and then subtracting (2) from (1),
`ll' tan theta +ml'sec theta - ll' tantheta + lm' sectheta = nl'-ln'`
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