# Question #d73e5

Feb 3, 2018

See explanation.

#### Explanation:

Well, let's see what we can do starting from the left-hand-side.
I see I can transform the $\tan x$ the following way:
$\sin x \tan x = \sin x \frac{\sin x}{\cos x} = \frac{{\sin}^{2} x}{\cos x}$
Now, we see we're getting closer since we already have the $\frac{1}{\cos} x$ part.
Let's now transform ${\sin}^{2} x$.
We recall the identity ${\sin}^{2} x + {\cos}^{2} x = 1$.
This becomes ${\sin}^{2} x = 1 - {\cos}^{2} x$. So we can now rewrite our equation:
$\sin x \tan x = \frac{{\sin}^{2} x}{\cos x} = \frac{1 - {\cos}^{2} x}{\cos x}$
Simplifying, we have:
$\sin x \tan x = \frac{1}{\cos x} - \frac{{\cos}^{2} x}{\cos x}$
and cancelling out the $\cos x$ in the rightmost term, we get:
$\sin x \tan x = \frac{1}{\cos x} - \cos x$
Q.E.D.