1.

If `tan theta +sin theta =m` and `tan theta -sin theta =n` then prove `m^2-n^2=4sqrt(mn)`

Answer» `m^2 = tan^2 theta + sin^2 theta + 2 tan theta sin theta`
`n^2 = tan^2 theta + sin^2 theta - 2 tan theta sin theta`
so, `m^2 - n^2 = 4 tan theta sin theta`
`4 sqrt(mn) = 4sqrt((tan theta + sin theta)(tan theta - sin theta))`
`= 4 sqrt(tan^2 theta - sin^2 theta)`
`= 4 sqrt(sin^2 theta/cos^2 theta - sin^2 theta `
`= 4 sin theta sqrt( sec^2 theta - 1) `
`4 sqrt(mn) = 4 sin theta tan theta `
`= m^2 - n^2 `
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