1. Powers of Cosine Express f(x) = cos 5x on 0 x pi as a linear combination of the functions K5 =
{cosj x}j=0 rightarrow 5 .
Solution
cos5x
= cosxcos4x - sinxsin4x
= cosx(2cos 2
2x -1) - 2sinxsin2xcos2x
=cosx(4cos 2
x -2 -1) - 4sin 2
xcosx(2cos 2
x - 1)
=4cos 3
x - 3cosx - 4(2cos 3
x - cosx)(1-cos 2
x)
=4cos 3
x - 3cosx - 4(2cos 3
x - 2cos 5
x - cosx + cos 3
x)
=4cos 3
x - 3cosx - 8cos 3
x + 8cos 5
x + 4cosx - 4cos 3
x
=8cos 5
x- 8cos 3
x + cosx
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