यदि `cos theta-sin theta=m` व `cos theta+sin theta=n.` तब सिद्ध कीजिए कि
`(m^(2)-n^(2))/(m^(2)+n^(2))=-2sin theta*cos theta=-(2)/(tan theta+cot theta)*`

Explanation

यहाँ, `cos theta+sin theta=m`
`implies(cos theta-sin theta)^(2)=m^(2)implies1-2cos thetasin theta=m^(2)`
`impliescos ^(2)theta+sin ^(2)theta-2cos thetasin theta=m^(2)" "...(1)`
अब, `cos theta+sin theta=nimplies(cos theta+sin theta)^(2)=n^(2)`
`impliescos ^(2)theta+sin ^(2)theta+2sin thetacos theta=n^(2)`
`implies1+2sin thetacos theta=n^(2)" "...(2)`
समीकरण (1 ) व (2 ) को जोड़ने पर
`2=m^(2)+n^(2)`
समीकरण (1 ) में से समीकरण (2 ) को घटाने पर,
`-4cos thetasin theta=m^(2)-n^(2)" "...(3)`
अब, `(m^(2)-n^(2))/(m^(2)+n^(2))=(-4cos thetasin theta)/(2)" "...(4)`
`implies(m^(2)-n^(2))/(m^(2)+n^(2))=-2sinthetacos theta`
क्योकि, `-2sin thetacos theta=(-2sin thetacos theta)/(1)" "...(5)`
`implies-2sin thetacos theta=(-2sin thetacos theta)/(sin ^(2)theta+cos ^(2)theta)=((-2sin thetacos theta)/(sin thetacos theta))/((sin ^(2)theta+cos ^(2)theta)/(sin thetacos theta))`
`implies-sin thetacos theta=(-2)/((sin theta)/(cos theta)+(cos theta)/(sin theta)+(cos ^(2)theta)/(sin thetacos theta))`
`implies-2sin theta cos theta=(-2)/((sin theta)/(cos theta)+(cos theta)/(sin theta))=-(2)/(tan theta+cot theta)" "...(6)`
समीकरण (5 ) व (6 ) से,
`(m^(2)-n^(2))/(m^(2)+n^(2))=-sin thetacos theta=(-2)/(tan theta+cot theta)*`