Simplify tan^-1[3sin2α/(5+3 cos⁡〖2 α〗 )] + tan-1[tan⁡α/4] , where -π/2<α<π/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`