Question

Prove the following trigonometric identities: `sqrt((1-sintheta)/(1+sintheta))=sectheta-tantheta` `sqrt((1+costheta)/(1-costheta))=cos e ctheta+cottheta`

Answer 1

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