Question

If m = ` ( cos theta - sin theta ) and n = ( cos theta +  sin theta ) "then show that" sqrt(m/n) + sqrt(n/m) = 2/sqrt(1-tan^2 theta)`.

Answer 1

LBS = `sqrt(m/n) + sqrt(n/m)`

       =`sqrt(m)/sqrt(n) + sqrt(m)/sqrt(n)`

       =`(m+n)/sqrt(mn)`

       =`((cos theta - sin theta ) + ( cos theta + sin theta ))/sqrt(( cos theta - sin theta ) ( cos theta + sin theta ))`

      =`(2 cos theta )/ sqrt( cos ^2 theta - sin^2 theta)`

      =`(2 cos theta ) / sqrt( cos ^ theta - sin^ theta)`

     =` ((( 2 cos theta )/( cos theta)))/((sqrt(cos^2 theta - sin^2 theta)/(cos theta))`

     =`2/(sqrt((cos^2 theta)/(cos^2 theta) - ( sin^2 theta) /( cos^2 theta))`

     =`2/sqrt(1- tan^2 theta)`

   = RHS