Question

Prove that
` sqrt( sec^(2) theta + "cosec"^(2) theta )= tan theta + cot theta . `

Answer 1

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