Question

How do you simplify `csc θsec θ - cot θ`?

Answer 1

`tan theta`

Explanation:

Lets start by using some trig identities to put the original statement into terms of `sin` and `cos`. Substituting the trig identities for `sec`, `csc`, and `cot` the expression becomes;

`1/sin theta 1/cos theta - cos theta/sin theta`

At this point it would be helpful to have a common denominator. We can do that by multiplying the top and bottom of the second term by `cos theta`.

`1/sin theta 1/cos theta - cos theta/sin theta cos theta/ cos theta`

Now we can combine the numerators over `sin theta cos theta`.

`(1 - cos^2 theta)/(sin theta cos theta)`

We can use the Pythagorean theorem to simplify the numerator. The Pythagorean theorem states that;

`sin^2 theta + cos^2 theta = 1`

Rearranging the terms we get;

`sin^2 theta = 1-cos^2 theta`

Now we can make the substitution for the numerator.

`sin^2 theta/(sin theta cos theta)`

The `sin theta` terms cancel, leaving;

`sin theta/ cos theta`

Which is the definition of `tan`.

`tan theta`