Prove that:

$cos 3 straight x space equals space 4 cos cubed straight x minus 3 cosx$

Solution

Consider cos3x = cos(x + 2x)
= cosx cos2x - sinx sin2x
$equals cosx open curly brackets 2 space cos squared straight x minus 1 close curly brackets space minus space sinx open curly brackets 2 space sinx space cosx close curly brackets$
$equals 2 space cos cubed straight x space minus space cosx space minus space 2 space sin squared straight x space cosx$
$equals 2 space cos cubed straight x minus cosx minus 2 left parenthesis 1 minus cos squared straight x right parenthesis cosx$
$equals 2 space cos cubed straight x minus cosx minus 2 cosx plus 2 cos cubed straight x$
$equals 4 space cos cubed straight x minus 3 space cosx$
∴ $space space space space space space space cos 3 straight x space equals space 4 space cos cubed straight x space minus space 3 space cosx$

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Principle Of Mathematical Induction

Prove that:

$cos 3 straight x space equals space 4 cos cubed straight x minus 3 cosx$

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