1. de Moivre question

I need help with this question:
By applying de Moivre's theorem and by also expanding (cos a + i sin a)^5, express cos 5a as a polynomial in cos a

Nothing jumps out, and for there must be an easier way that expanding the whole thing. Plus I don't fully understand the question

Thanks

2. You can say,
$(cos x+i\sin x)^5=\cos 5x+i\sin 5x$
Also, you can expand this,
$a=\cos x$ and $b=i\sin x$
Then,
$(a+b)^5=\sum_{k=0}^5{5\choose k}a^kb^{n-k}$
By binomal formula.

Just so you remember,
$i^2=-1$
$i^3=-i$
$i^4=1$
....

3. ~ ~ ~

Hello, chancey!

DeMoivre's Theorem: . $(\cos\theta + i\sin\theta)^n\;=\;\cos(n\theta) + i\sin(n\theta)$

By applying DeMoivre's theorem and by also expanding $(\cos\theta + i\sin\theta)^5$,
express $\cos5\theta$ as a polynomial in $\cos\theta$.

From the theorem, we have: . $\cos5\theta + i\sin5\theta \;= \;(\cos\theta + i\sin\theta)^5$

Expand the right side:
. $\cos5\theta + i\sin5\theta \;= \;\cos^5\!\theta + 5i\cos^4\!\sin\theta - 10\cos^3\!\theta\sin^2\!\theta$ $- 10i\cos^2\!\theta\sin^3\!\theta + 5\cos\theta\sin^4\!\theta + i\sin^5\!\theta$

And we have: . $\cos5\theta + i\sin5\theta$
. . $= \;\left(\cos^5\!\theta - 10\cos^3\!\theta\sin^2\!\theta + 5\cos\theta\sin^4\!\theta\right)$ $+ i\left(5\cos^4\!\theta\sin\theta - 10\cos^2\!\theta\sin^3\!\theta + \sin^5\!\theta\right)$

Equate real components: . $\cos5\theta\;= \;\cos^5\!\theta - 10\cos^3\!\theta\sin^2\!\theta + 5\cos\theta\sin^4\!\theta$ **

Replace $\sin^2\!\theta$ with $1 - \cos^2\!\theta:$
. . $\cos5\theta \;= \;\cos^5\!\theta - 10\cos^3\!\theta\left(1 - \cos^2\!\theta\right) + 5\cos\theta\left(1 - \cos^2\!\theta\right)^2$

Therefore: . $\boxed{\cos5\theta \;= \;16\cos^5\!\theta - 20\cos^3\!\theta + 5\cos\!\theta}$

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** .If we equate the 'imaginary' components, we have:
. . . . $\sin5\theta \;= \;5\cos^4\!\theta\sin\theta - 10\cos^2\!\theta\sin^3\!\theta + \sin^5\!\theta$

. . which simplifies to: . $\boxed{\sin5\theta \;= \;5\sin\theta - 20\sin^3\!\theta + 16\sin^5\!\theta}$

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Another bonus . . .

$\tan5\theta \;= \;\frac{\sin5\theta}{\cos5\theta} \;= \;\frac{5\cos^4\!\theta\sin\theta - 10\cos^2\!\theta\sin^3\!\theta + \sin^5\!\theta}{\cos^5\!\theta - 10\cos^3\!\theta\sin^2\!\theta + 5\cos\theta\sin^4\!\theta}$

Divide top and bottom by $\cos^5\!\theta:$

$\tan5\theta \;= \;\frac{\frac{5\cos^4\!\theta\sin\theta}{\cos^5\!\ theta} - \frac{10\cos^2\!\theta\sin^3\!\theta}{\cos^5\!\the ta} + \frac{\sin^5\!\theta}{\cos^5\!\theta}}{\frac{\cos^ 5\!\theta}{\cos^5\!\theta} - \frac{10\cos^3\!\theta\sin^2\!\theta}{\cos^5\!\the ta} + \frac{5\cos\theta\sin^4\!\theta}{\cos^5\!\theta} }$ . $= \;\frac{5\frac{\sin\theta}{\cos\theta} - 10\frac{\sin^3\!\theta}{\cos^3\!\theta} + \frac{\sin^5\!\theta}{\cos^5\!\theta}}{1 - 10\frac{\sin^2\!\theta}{\cos^2\!\theta} + 5\frac{\sin^4\!\theta}{\cos^4\!\theta} }$

Therefore: . $\boxed{\tan5\theta \;= \;\frac{5\tan\theta - 10\tan^3\!\theta + \tan^5\!\theta}{1 - 10\tan^2\!\theta + 5\tan^4\!\theta}}$