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$Line = 0;
“Find an identity for cos5x in terms of cosx.”
-----------------------------------TrigExpand@exprD, “expands out trigonometric functions in expr. “
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Cos@5 xD  TrigExpand
Cos@xD5 - 10 Cos@xD3 Sin@xD2 + 5 Cos@xD Sin@xD4
Relace Sin@xD2 with H1 - HCos@xD ^ 2LL and Sin@xD4 with H1 - HCos@xD ^ 2LL2 . Then multiply out, “Expand”, the resulting expression. Simplification is performed automatically.
2
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%1 . 9Sin@xD2 ® I1 - Cos@xD2 M, Sin@xD4 ® I1 - Cos@xD2 M =  Expand
5 Cos@xD - 20 Cos@xD3 + 16 Cos@xD5
Equate Cos@xD5 to the result above and test the equation with x = Π/6. True indicates that the LHS equals the RHS.
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ICos@5 xD == 5 Cos@xD - 20 Cos@xD3 + 16 Cos@xD5 M . x ® Π  6
True