Simplify `[(3sin2alpha)/(5+3cos2alpha)]+tan^(-1)[(tanalpha)/4],w h e r e-pi/2

Explanation

`tan^(-1) [(3 sin 2 alpha)/(5 + 3 cos 2 alpha)] + tan^(-1) [(tan alpha)/(4)]`
`=tan^(-1) ((4 tan alpha)/(8 + 2 tan^(2) alpha)) + tan^(-1) ((tan alpha)/(4))`
`= tan^(-1). (((3tan alpha)/(4 + tan^(2) alpha) + (tan alpha)/(4))/(1- (3 tan^(2) alpha)/(16 + 4 tan^(2) alpha))) [ :' (3 tan^(2) alpha)/(16 + 4 tan^(2) alpha) lt 1]`
`= tan^(-1) ((12 tan alpha + 4 tan alpha + tan^(3) alpha)/(16 + tan^(2) alpha))`
`= tan^(-1) (tan alpha) = alpha`