<s>[INST] Proving the given inequality. (Complex Analysis). We're given a function $P_n(x)$ for $-1\leq x\leq1$ as follows :
$$P_n(x) = \int \limits_0^\pi \dfrac{1}{\pi}(x+i\sqrt{1-x^2} \cos\theta)^n \, d\theta$$
for $n=(0,1,2,3,\ldots)$, we need to prove that $|P_n(x)| \leq 1$.
I tried the following :
Let $z=x+i\sqrt{1-x^2}\cos\theta$ 
I somehow want to prove that $|z|=|x+i\sqrt{1-x^2}\cos\theta|\leq1$, as that would imply that $|z^n|\leq1$, as $|z^n|=|z|^n$.
$$|z| = \sqrt{x^2+(1-x^2)\cos^2 \theta} \Longrightarrow \sqrt{x^2 \sin^2\theta + \cos^2\theta}.$$
Now, $x^2\leq1$ and $\sin^2\theta \leq1 \Longrightarrow x^2\sin^2\theta \leq1$.
Also, $\cos^2\theta \leq1 \Longrightarrow \sqrt{x^2\sin^2\theta+\cos^2\theta} \leq \sqrt{2}$, 
But " $\leq1$" condition is required...
That's the point where I am stuck, could anyone help me with this?
 [/INST] HINT: $(x^2+(1-x^2) \cos^2\theta) \leq (x^2+(1-x^2))=1$, because $\theta$ is real.
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