I have to answer the following and support it with a full proof.
Is cos(5°) rational? (Given hint: show that if cos(x) is rational, then cos(3x) is rational).
$\displaystyle \cos (3x)=\cos (x+2x)$
$\displaystyle \cos x\cos 2x-\sin x\sin 2x$
$\displaystyle \cos x(\cos ^2 x-\sin^2 x)-\sin x (2\sin x\cos x)$
$\displaystyle \cos^3 x-3\cos x \sin^2 x$
And,
$\displaystyle \sin^2 x=1-\cos ^2x$
So if $\displaystyle a=\cos x $ is rational
Then,
$\displaystyle a^3-3 a(1-a^2)$ is also rational.
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If $\displaystyle \cos (5^o)$ is rational.
Then,
$\displaystyle \cos (15^o)$ is rational.
Then,
$\displaystyle \cos (45^o)=\frac{\sqrt{2}}{2}$ is rational.
But that is not true.
Contradiction.
Yes it does.
If assume $\displaystyle \cos 5^o$
Is rational and you arrived at a contradiction.
If $\displaystyle a=\cos x$ is a rational number then,
$\displaystyle \cos 3x=a^3-3a(1-a^2)$ is a rational number.
Because when you add or multiply rationals you get a rational.