<s>[INST] Does $\sqrt{\cos{\theta} \sqrt{\cos{\theta} \sqrt{\cos{\theta\dots}}}}=\sqrt{1 \sqrt{1 \sqrt{1\dots}}} \implies \cos{\theta}=1$? I was solving this equation:
$$\sqrt{\cos{\theta} \sqrt{\cos{\theta} \sqrt{\cos{\theta\dots}}}}=1$$
I solved it like this:
The given equation can be written as:
\begin{align*}
\sqrt{\cos{\theta} \sqrt{\cos{\theta} \sqrt{\cos{\theta\dots}}}}&=\sqrt{1 \sqrt{1 \sqrt{1\dots}}} \\
\cos{\theta}&=1 \\
\theta&=\arccos {1}
\end{align*}
So the solution is $2n\pi, n \in \mathbb Z$.
Have I solved it the wrong way?
(The title originally contained a more general question: Does $\sqrt{a \sqrt{a \sqrt{a\dots}}}=\sqrt{b \sqrt{b \sqrt{b\dots}}} \implies a=b$? The current title is consistent with the body and the accepted answer.)
 [/INST] Hint:
$$\sqrt{\cos{\theta} \sqrt{\cos{\theta} \sqrt{\cos{\theta}}}}...=1.$$ 
Multiply by $\cos(\theta)$ and take the square root. You get
$$\sqrt{\cos(\theta)}=\sqrt{\cos{\theta} \sqrt{\cos{\theta} \sqrt{\cos{\theta} \sqrt{\cos{\theta}}}}}...=1.$$ 
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