Question

Prove that
`(i) (1- sin theta)/(1+ sin theta)=(sec theta - tan theta )^(2) `
`(ii) ((1+cos theta))/((1- cos theta))= ("cosec" theta + cot theta)^(2)`

Answer 1

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