यदि `cosec theta-sin theta=l` तथा `sec theta-cos theta =m,` तो सिद्ध करो कि
`l^(2)m^(2)(l^(2)+m^(2)+3)=1`

Explanation

`LHS=l^(2)m^(2)(l^(2)+m^(2)+3)" "` [l तथा m के मान रखने पर]
`impliesLHS =(cosec theta -sin theta)^(2)(sec theta -cos theta )^(2){(cosec theta-sin theta )^(2)+(sec theta -cos theta )^(2)+3}`
`impliesLHS =((1)/(sin theta )-sin theta )^(2)((1)/(cos theta )-cos theta)^(2){((1)/(sintheta)-sin theta )^(2)+((1)/(cos theta )-cos theta )^(2)+3}`
`impliesLHS=((1-sin ^(2)theta )/(sin theta ))^(2)((1-cos ^(2)theta )/(cos theta)){((1-sin^(2)theta)/(sin theta))^(2)+((1-cos ^(2)theta)/(cos theta))^(2)+3}`
`impliesLHS =((cos ^(2)theta)/(sin theta))^(2)((sin ^(2)theta )/(cos theta))^(2){((cos ^(2)theta )/(sin theta))^(2)+((sin ^(2)theta)/(cos theta))^(2)+3}`
`impliesLHS =(cos ^(4)theta )/(sin ^(2)theta)xx(sin ^(4)theta)/(cos ^(2)theta){(cos ^(4)theta)/(sin ^(2)theta)+(sin ^(4)theta )/(cos ^(2)theta)+3}`
`impliesLHS =cos ^(2)theta xxsin ^(2)theta {(cos ^(6)theta+sin ^(6)theta+ 3 cos ^(2)thetasin ^(2)theta)/(cos ^(2)theta sin ^(2)theta )}`
`impliesLHS =cos ^(6)theta+sin ^(6)theta +3 cos ^(2)theta sin ^(2)theta`
`impliesLHS ={(cos ^(2)theta )^(3)+(sin ^(2)theta)^(3)}+3 cos ^(2)theta sin ^(2)theta`
`impliesLHS ={(cos ^(2)theta +sin^(2)theta)^(3)-3 cos ^(2)theta sin ^(2)theta (cos ^(2)theta + sin ^(2)theta)}+3sin^(2) theta cos ^(2)theta`
`[because a^(3)+b^(3)=(a+b)^(3)-3ab(a+b)]`
`impliesLHS ={1-3 cos ^(2)thetasin ^(2)theta }+3cos ^(2) theta sin^(2)theta `
`implies LHS =1=RHS`