<s>[INST] Proving Trig Identities (Complex Numbers) 

Question: Prove that if $z = \cos (\theta) + i\sin(\theta)$, then
$$ z^n +  {1\over z^n} = 2\cos(n\theta) $$
Hence prove that $\cos^6(\theta)$ $=$ $\frac 1{32}$$(\cos(6\theta)$ + $6\cos(4\theta)$ + $15\cos(2\theta)$ + $10$)


I learnt to prove the first part in another post linked here.
The second part is where I am confused because there is a 'hence'
so I thought of taking 2 approaches:
either $$ z^6 + \frac 1{z^6} $$
or $$ z^6 $$ and equating real parts.
I will start with my first approach 
$$ z^6 + \frac 1{z^6} $$
$$ (\cos(x)+i\sin(x))^6 + \frac 1{(\cos(x)+i\sin(x))^6} $$
$$ (\cos(x)+i\sin(x))^6 + {(\cos(x)+i\sin(x))^{-6}} $$
$$ \cos(6x) + i\sin(6x) + \cos(-6x)+ i \sin(-6x) $$
$$ 2\cos(6x) $$ 
Which is no where near what I am suppose to prove..
So with my second approach (expanding and equation real parts)
$$ z^6 $$ 
$$ (\cos(x) + i \sin(x))^6 $$
Using pascals 
$$ \cos^6(x) + i*6\cos^5(x)\sin(x) + i^2*15\cos^4(x)\sin^2(x) + i^3*20\cos^3(x)\sin(x) + i^4*15\cos^2(x)\sin^4(x) + i^5*6\cos(x)\sin^5(x) + i^6 * \sin^6(x) $$
Simplifying
$$ \cos^6(x) - 15\cos^4(x)\sin^2(x)+ 15\cos^2(x)\sin^4(x) - \sin^6(x) +i(6\cos^5(x)\sin(x)-20\cos^3(x)\sin(x) + 6cos(x)\sin^5(x)$$
Now considering only real 
$$ \cos^6(x) - 15\cos^4(x)\sin^2(x)+ 15\cos^2(x)\sin^4(x) - \sin^6(x) $$
At this point I'm confused , am I on the right approach?
 [/INST] By De Moivre's formula if $z = \cos (\theta) + i\sin(\theta)$ then $z^n = \cos (n\theta) + i\sin(n\theta)$.
So $$z^n = \cos (n\theta) + i\sin(n\theta)$$ 
and 
$$z^{-n} = \cos (-n\theta) + i\sin(-n\theta)=\cos (n\theta) - i\sin(n\theta)$$
So $$z^n+z^{-n}=2\cos (n\theta)$$
Thus when $n=1$ , 
$$z+z^{-1}=2\cos (\theta) \Rightarrow \left(z+\frac{1}{z} \right)^6=2^6 \cdot \cos^6(\theta) $$ 
$$2^6 \cdot \cos^6(\theta)=\left(z+\frac{1}{z} \right)^6 $$ 
$$2^6 \cdot \cos^6(\theta)=z^6+6z^4+15z^2+20+15+\frac{1}{z^2} +6\frac{1}{z^4} +\frac{1}{z^6}  $$ 
$$2^6 \cdot \cos^6(\theta)=\left(z^6+\frac{1}{z^6}\right)+6\left(z^4+\frac{1}{z^4}\right)+15\left(z^2+\frac{1}{z^2}\right)+20 $$
$$64 \cdot \cos^6(\theta)=\left(2\cos 6\theta \right)+6\left(2\cos 4\theta\right)+15\left(2\cos 2\theta\right)+20 $$
Therefore 
$$  \cos^6(\theta)=\frac{1}{32}\{\left(\cos 6\theta \right)+6\left(\cos 4\theta\right)+15\left(\cos 2\theta\right)+10 \}$$
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