<s>[INST] Why is $\frac{0.5}{ \cos^2(30)} = \frac{\tan(30)}{\cos(30)} $, but $\frac{0.5}{ \cos^2(13) } \neq \frac{\tan(13)}{\cos(13)} $? Is just it a coincidence that
$$\frac{0.5}{ \cos^2(30)} = \frac{\tan(30)}{\cos(30)} $$
However
$$\frac{0.5}{ \cos^2(13) } \neq \frac{\tan(13)}{\cos(13)} $$
is not equal ? And if not does anyone know a reason why they just so happen to be equatable?
 [/INST] If we use $x$ as a variable, we get the following:
$$\begin{align}
& \frac {0.5}{\cos^2x} = \frac {\tan x}{\cos x} \\
\implies& \frac {1}{2} \cos x = \cos^2 x \tan x\\
\implies& \cos x \tan x\ = \frac {1}{2} \\
\implies& \cos x \frac {\sin x}{\cos x} = \frac {1}{2}\\
\implies&  \sin x\ = \frac {1}{2}\
\end{align}$$
Thus the only solutions to the above equation are $x = 30^\circ \pm 360^\circ k$ or $x = 150^\circ \pm 360^\circ k$ where $k$ is any natural number.
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