Question

`SinA + 2Sin2A + Sin3A` is equal to which of the following? (1.) `4sin2A cos^2(A/2) (2.) 2sin2A(sinA/2+cosA/2)^2 (3.) 8sinAcosAcos^2(A/2)`

Answer 1

`sinA+2sin2A+sin3A = (sinA+sin3A)+2sin2A`
`=2sin2AcosA+2sin2A ` (Since `sinX+sinY = 2sin((X+Y)/2)cos((X-Y)/2)`)
`= 2sin2A(1+cosA)`
`=2sin2A(2cos^2(A/2))`
`=4sin2Acos^2(A/2)`
So, option `(1)` is correct option.
Also, we can write,
`4sin2Acos^2(A/2) = 4(2sinAcosA)cos^2(A/2) = 8sinAcosAcos^2(A/2)`
So, option `(3)` is also correct.