Question

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

Answer 1

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