Question: $\csc x = 4 \sec x$ within the range of $0^\circ-360^\circ$. I attempted to do this without using $\tan x$ - challenging myself.

My attempt: 
$$\begin{split}
\dfrac{1}{\sin x}&= \dfrac{4}{\cos x} \\
1&= \dfrac{4 \sin x}{\cos x} \\
\cos x &= 4 \sin x \\
1- \sin x&= 4 \sin x \\
5 \sin x &= 1 \sin x = 1/5 \\
&\arcsin(1/5)
\end{split}
$$
This gave me the wrong answer. Is it because $\cos x = 1-\sin x$ is wrong? Is it only possible if sine and cosine are squared? I hope some one can clear this up. Thank you.