Question 474432
  <pre><font color = "indigo"  size=4><b>
It comes from the fact that 

sin(n<font face="symbol">p</font>) = 0

and

cos(0<font face="symbol">p</font>) = 1, cos(1<font face="symbol">p</font>) = -1, cos(2<font face="symbol">p</font>) = 1, cos(3<font face="symbol">p</font>) = -1, cos(4<font face="symbol">p</font>) = 1, etc.

cos([even number]·<font face="symbol">p</font>) = 1,  cos([odd number]·<font face="symbol">p</font>) = -1 

and

(-1)<sup>(even number)</sup> = 1,  (-1)<sup>(odd number)</sup> = -1

therefore, since even and odd numbers alternate, we have 

cos(n<font face="symbol">p</font>) = (-1)<sup>n</sup>

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Therefore,

sin(n<font face="symbol">p</font> + <font face="symbol">q</font>) = 

sin(n<font face="symbol">p</font>)cos(<font face="symbol">q</font>) + cos(n<font face="symbol">p</font>)sin(<font face="symbol">q</font>) =

     0·cos(<font face="symbol">q</font>) + (-1)<sup>n</sup>sin(<font face="symbol">q</font>) =

                  (-1)<sup>n</sup>sin(<font face="symbol">q</font>) 

Edwin</pre>