# How do you simplify tan x cos x csc x?

Jul 22, 2015

It equals 1 for all values of $x$ where each factor is defined.

#### Explanation:

Using the definition of tangent and cosecant gives:

$\tan \left(x\right) \cos \left(x\right) \csc \left(x\right) = \frac{\sin \left(x\right)}{\cos \left(x\right)} \cdot \cos \left(x\right) \cdot \frac{1}{\sin \left(x\right)}$

Everything now cancels to give

$\tan \left(x\right) \cos \left(x\right) \csc \left(x\right) = \frac{\cancel{\sin \left(x\right)}}{\cancel{\cos \left(x\right)}} \cdot \cancel{\cos \left(x\right)} \cdot \frac{1}{\cancel{\sin \left(x\right)}} = 1$

for all values of $x$ where each of the original factors is defined.

The values of $x$ where this is not true are those values of $x$ which make either $\cos \left(x\right) = 0$ or $\sin \left(x\right) = 0$. One of these will happen at each value of $x$ that is an integer multiple of $\frac{\pi}{2}$ radians (90 degrees).

Hence, $\tan \left(x\right) \cos \left(x\right) \csc \left(x\right) = 1$ for all $x$ except $x = \frac{n \pi}{2}$, where $n = 0 , \setminus \pm 1 , \setminus \pm 2 , \setminus \pm 3 , \ldots$