TITLE: Does $\sqrt{\cos{\theta} \sqrt{\cos{\theta} \sqrt{\cos{\theta\dots}}}}=\sqrt{1 \sqrt{1 \sqrt{1\dots}}} \implies \cos{\theta}=1$?
QUESTION [7 upvotes]: 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.)

REPLY [3 votes]: Another answer
suppose that $$\sqrt{\cos{\theta} \sqrt{\cos{\theta} \sqrt{\cos{\theta}}}}...=1$$
square both side you get $$cos{\theta} \sqrt{\cos{\theta} \sqrt{\cos{\theta}}}...=1$$
We know that $$\ \sqrt{\cos{\theta} \sqrt{\cos{\theta}}}...=1$$
thus $$\cos{\theta}*1=1$$
therefore $$\theta=2k\pi,k\in\mathbb Z$$