Prove that $cos 3 x = 4 {cos}^{3} x - 3 cos x$

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Solution

To prove :
$c o s 3 x = 4 c o s^{3} x - 3 c o s x$
Proof :
$c o s 3 x = c o s (2 x + x)$
$= c o s 2 x c o s x - s i n 2 x s i n x$
$= (- 1 + 2 c o s^{2} x) c o s x - 2 c o s x s i n x s i n x$
$= - c o s x + 2 c o s^{3} x - 2 s i n^{2} x c o s x$
$= - c o s x + 2 c o s^{3} x - 2 (1 - c o s^{2} x) c o s x$
$= - 3 c o s x + 4 c o s^{3} x$
Hence, $c o s 3 x = 4 c o s^{3} x - 3 c o s x$

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