सिद्ध कीजिए - `m^(2)-n^(2)=4sqrt(m.n)`, यदि `tan theta + sin theta = m` एवं `tan theta - sin theta = n`.

Explanation

`because L.H.S. =m^(2)-n^(2)`
`=(tan theta + sin theta)^(2)-(tan theta - sin theta)^(2)`
`= (tan theta + sin theta + tan theta - sin theta) xx (tan theta + sin theta - tan theta + sin theta) " " [because x^(2)-y^(2)=(x+y)(x-y)" से"]`
`rArr m^(2)-n^(2)=2 tan theta xx 2 sin theta`
`= 4 tan theta sin theta " "` ....(1)
अब `because R.H.S. = 4sqrt(m.n)`
`=4sqrt((tan theta + sin theta)(tan theta - sin theta))`
`= 4sqrt(tan^(2)theta - sin^(2)theta)`
`rArr R.H.S. = 4sqrt((sin^(2)theta)/(cos^(2)theta)-sin^(2)theta)=4sqrt(sin^(2)theta((1)/(cos^(2)theta)-1))`
`= 4sqrt(sin^(2)theta(sec^(2)theta-1))=4sqrt(sin^(2)theta tan^(2)theta)`
`rArr 4sqrt(m.n)=4 tan theta sin theta " "` ...(2)
`rArr m^(2)-n^(2)=4sqrt(m.n)" "` [सेमी. (1) एवं (2) से]
`rArr L.H.S. = R.H.S. " "` इति सिद्धम्