<s>[INST] A trigonometric equation - Finding the value of theta I have to find the value of $\theta$ in this equation "$\cos2\theta \sin\theta = 1$"
The problem is that I have to solve using algebra (making equations...etc) like this one, for example: \begin{align} 4\sin^2\theta - 1 = 0\\ \sin^2\theta = 0.25\\ \sin\theta = +0.5, -0.5\\ \theta = 30, 150, 210, 330\end{align}
But for "$\cos2\theta \sin\theta = 1$", I figured, according to the fact that $\cos\alpha$ can be only between -1 and 1. And the same for $\sin\alpha$, that the only two expected solutions are when $\cos2\theta$ and $\sin\theta$ are both equal to 1 or -1 so that the product is equal to 1 as in the main equation.
So $\theta = 270$. \begin{align} \cos(540)\sin(270) = 1\\ \cos(180)\sin(270) = 1\\ -1 * -1 = 1 \end{align}
I'm trying to find another way to solve it like the way I solved the other example $4\sin^2\theta - 1 = 0$..
 [/INST] Hint:
Use the identity-$\cos2\theta=\cos^2\theta-\sin^2\theta$
So,$$\cos2\theta\sin\theta=1$$
$$\implies(\cos^2\theta-\sin^2\theta)\sin\theta=1$$
Now,replace $\cos^2\theta$ by $1-\sin^2\theta$
So,$$(1-2\sin^2\theta)\sin\theta=1$$
$$\implies\sin\theta-2\sin^3\theta-1=0$$
Now,let,$\sin\theta=x$.So,this reduces to a cubic equation.Now,solve for $\sin\theta$ and just plug the value in the equation.
Hope this helps!!
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