Fun Spesh trig question

[VCE_2012]

Expand cos(10x) and express it in terms of cos(x) and/or sin(x).

[pi]

http://mathworld.wolfram.com/Multiple-AngleFormulas.html

[brightsky]

let z = cis^(10)(x) = cis(10x)
Re(z) = cos(10x)
but z = (cos(x) + i sin(x))^10 = sum^(10)_(r = 0) 10 C r * (cos(x))^(10 - r) (i sin(x))^r
remembering that i^2 = -1, i^3 = -i, i^4 = 1, i^5 = i
we get real terms when r = 0, 2, 4, 6, 8, 10
the sum of these terms is equal to cos(10x)