Prove that `sinA+sin2A+sin4A+sin5A=4 (cosA/2)(cos3A/2)sin3A`

Explanation

We have to proof,

`sin(A^o)+sin(2A^o)+sin(4A^o)+sin(5A^o)=4 (cosA/2)(cos3A/2)sin3A`

Let,

L.H.S.=`sin(A^o)+sin(2A^o)+sin(4A^o)+sin(5A^o)`

As we know the formula,
`sin(A^o)+sin(B^o)= 2[sin(((A^o) +(B^o))/2)cos(((A^o)-(B^o))/2)]`

then, L.H.S.=`sin(A^o)+sin(2A^o)+sin(4A^o)+sin(5A^o)`

L.H.S=`2[(sin(((A^o)+(2A^o))/2)cos(((A^o)-(2A^o))/2)]+2[(sin(((4A^o)+(5A^o))/2)cos(((4A^o)-(5A^o))/2)]`

L.H.S=`2sin((3/2)A^o)cos((1/2)A^o)+2sin((9/2)A^o)cos((1/2)A^o)`

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