# How do you verify sinx/cosx + cosx/sinx = 1?

Feb 10, 2016

You can't verify it since it is not an identity.

#### Explanation:

You can't since this is not true.

To prove that this is not an identity, find one $x$ for which this equation is not true.

For example, you can take $x = \frac{\pi}{3}$:

As you know, $\sin \left(\frac{\pi}{3}\right) = \frac{\sqrt{3}}{2}$ and $\cos \left(\frac{\pi}{3}\right) = \frac{1}{2}$.

$\sin \frac{\frac{\pi}{3}}{\cos} \left(\frac{\pi}{3}\right) + \cos \frac{\frac{\pi}{3}}{\sin} \left(\frac{\pi}{3}\right) = \frac{\frac{\sqrt{3}}{2}}{\frac{1}{2}} + \frac{\frac{1}{2}}{\frac{\sqrt{3}}{2}} = \frac{\sqrt{3}}{1} + \frac{1}{\sqrt{3}} = \frac{4}{\sqrt{3}} \ne 1$

Thus, this equation is not an identity.

Feb 10, 2016

The given equation is not true
and therefore can not be verified.

#### Explanation:

$\frac{\sin x}{\cos x} + \frac{\cos x}{\sin x} = \frac{1}{\sin \left(x\right) \cdot \cos \left(x\right)} \ne 1$

As an obvious counter-example
if $x = \frac{\pi}{4}$
$\textcolor{w h i t e}{\text{XXX}} \sin \left(\frac{\pi}{4}\right) = \cos \left(\frac{\pi}{4}\right)$

$\Rightarrow \frac{\sin \left(\frac{\pi}{4}\right)}{\cos \left(\frac{\pi}{4}\right)} + \frac{\cos \left(\frac{\pi}{4}\right)}{\sin \left(\frac{\pi}{4}\right)} = 1 + 1 = 2 \ne 1$