Question

Write the trigonometric expression in terms of sine and cosine, and then simplify. cot()/sin()-csc()

Answer 1

First, we know that:

cot(x) = cos(x)/sin(x)

csc(x) = 1/sin(x)

I can't know for sure what is the exact equation, so I will assume two cases.

The first case is if the equation is:

`(cot(x))/(sin(x)) - csc(x)`

if we replace cot(x) and csc(x) we get:

`(cot(x))/(sin(x)) - csc(x) = (cos(x))/(sin(x)) (1)/(sin(x)) - (1)/(sin(x))`

Now let's we can rewrite this as:

`(cos(x))/(sin(x)) (1)/(sin(x)) - (1)/(sin(x)) =(cos(x))/(sin^2(x)) - (1)/(sin(x))`

`(cos(x))/(sin^2(x)) - (sin(x))/(sin^2(x)) = (cos(x) - sin(x))/(sin^2(x))`

We can't simplify it more.

Second case:

If the initial equation was

`(cot(x))/(sin(x) - csc(x))`

Then if we replace cot(x) and csc(x)

`(cos(x))/(sin(x))*(1)/(sin(x) - 1/sin(x)) = (cos(x))/(sin(x))*(1)/(sin^2(x)/sin(x) - 1/sin(x))`

This is equal to:

`(cos(x))/(sin(x))*(sin(x))/(sin^2(x) - 1)`

And we know that:

sin^2(x) + cos^2(x) = 1

Then:

sin^2(x) - 1 = -cos^2(x)

So we can replace that in our equation:

`(cos(x))/(sin(x))*(sin(x))/(sin^2(x) - 1) = (cos(x))/(sin(x))*(sin(x))/(-cos^2(x)) = -(cos(x))/(cos^2(x))*(sin(x))/(sin(x)) = - (1)/(cos(x))`