Question

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

Answer 1

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

Explanation:

Using the definition of tangent and cosecant gives:

`tan(x)cos(x)csc(x)=(sin(x))/(cos(x))*cos(x)*1/(sin(x))`

Everything now cancels to give

`tan(x)cos(x)csc(x)=(cancel(sin(x)))/(cancel(cos(x)))*cancel(cos(x))*1/(cancel(sin(x)))=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(x)=0` or `sin(x)=0`. One of these will happen at each value of `x` that is an integer multiple of `pi/2` radians (90 degrees).

Hence, `tan(x)cos(x)csc(x)=1` for all `x` except `x=(n pi)/2`, where `n=0,\pm 1, \pm 2, \pm 3,...`