Question

To prove `(cosA/1+sinA)+(1+sinA/cosA) = 2 secA

Answer 1

`cos A = (cos^2 (A/2) - sin^2(A/2)) = (cos(A/2)+sin(A/2))(cos(A/2)-sin(A/2))`
`1+sinA = cos^2(A/2)+sin^2(A/2) +2sin(A/2)cos(A/2) = (cos(A/2)+sin(A/2))^2`
`:. cosA/(1+sinA) = (cos(A/2)-sin(A/2))/(cos(A/2)+sin(A/2))`
`:. L.H.S. = cosA/(1+sinA)+(1+sinA)/cosA =(cos(A/2)-sin(A/2))/(cos(A/2)+sin(A/2)) +(cos(A/2)+sin(A/2))/(cos(A/2)-sin(A/2)) `
`= ((cos(A/2)-sin(A/2))^2+(cos(A/2)+sin(A/2))^2)/(cos^2(A/2) - sin^2(A/2))`
`=(cos^2(A/2)+sin^2(A/2) +2sin(A/2)cos(A/2)+cos^2(A/2)+sin^2(A/2) -2sin(A/2)cos(A/2))/(cosA)`
`=(1+1)/cosA`....(As `cos^2(A/2)+sin^2(A/2) = 1`)
` = 2secA = R.H.S.`