Question

Prove that
`(1)/((sec theta - tan theta))-(1)/(cos theta)=(1)/(cos theta)-(1)/((sec theta+ tan theta)). `

Answer 1

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