Question

`sqrt((1+cos theta)/(1-cos theta)) + sqrt((1-cos theta )/(1+ cos theta )) = 2 cosec theta`

 

Answer 1

LHS=`sqrt((1+cos theta)/(1-cos theta)) + sqrt((1-cos theta )/(1+ cos theta ))`

    =`sqrt(((1+cos theta)^2)/((1-cos theta)(1+ cos theta))) + sqrt (((1-cos theta)^2)/((1+ cos theta) (1- cos theta))`

    =`sqrt(((1+cos theta)^2)/((1-cos^2 theta))) + sqrt(((1-cos theta )^2)/((1-cos^2 theta))`

    =` sqrt(((1+ cos theta)^2)/(sin^2 theta))+sqrt(((1-cos theta
)^2)/sin^2 theta)`

     =`((1+cos theta))/(sin theta) + ((1-cos theta))/(sin theta)`

      =`(1+ cos theta +1-cos theta)/sin theta`

     =`2/sin theta`

    = 2cos ecθ
   = RHS