How do you simplify tan x cos x csc x?

1 Answer
Bill K.
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(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,...

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