Question

Prove the following identities:

`("cosec"  theta + cot theta)/("cosec"  theta - cot theta) = ("cosec"  theta + cot theta )^2 = 1 + 2 cot^2 theta + 2  "cosec"  theta cot theta`

Answer 1

Here, `( cosec theta + cot theta )/( cosec theta - cot theta)`

        = `((cosec theta + cot theta) ( cosec theta + cot theta ))/(( cosec theta - cot theta ) ( cosec theta + cot theta))`

        =` ((cosec theta + cot theta)^2)/(( cosec ^2 theta - cot^2 theta))`

       =`((cosec theta + cot theta )^2) /1`

      =`(cosec theta + cot theta )^2`

 Again , `( cosec theta + cot theta )^2`

     = ` cosec^2 theta + cot^2 theta + 2 cosec theta  cot theta `

     =` 1+cot^2 theta + cot^2 theta + 2 cosec theta  cot theta (∵ cosec^2 theta - cot^2 theta =1)`

    =` 1+2 cot^2 theta + 2 cosec theta  cot theta `