Question

Prove that: `sqrt((sectheta-1)/(sectheta+1))+sqrt((sectheta+1)/(sectheta-1))=2 cos e c theta`

Answer 1

We have
`LHS = sqrt((sec theta -1)/(sec theta +1)) + sqrt((sec theta +1)/(sec theta -1))= (sqrt(sec theta -1))/(sqrt(sec theta +1)) + (sqrt(sec theta +1))/(sqrt(sec theta -1)) `
` = (sec theta -1+ sec theta +1)/(sqrt((sec theta +1)(sec theta -1))) =(2 sec theta)/(sqrt(sec^(2)theta -1))=(2sec theta)/(tan theta) " "[ because sec^(2)theta -1= tan^(2)theta] `
`= 2 sec theta cot theta = (2)/(cos theta) xx (cos theta)/(sin theta)=(2)/(sin theta) = 2 "cosec" theta = RHS.`
` therefore LHS = RHS. `