Question

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

Answer 1

Answer:

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:

$\frac{cot(x)}{sin(x)} - csc(x)$

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

$\frac{cot(x)}{sin(x)} - csc(x) = \frac{cos(x)}{sin(x)} \frac{1}{sin(x)} - \frac{1}{sin(x)}$

Now let's we can rewrite this as:

$\frac{cos(x)}{sin(x)} \frac{1}{sin(x)} - \frac{1}{sin(x)} =\frac{cos(x)}{sin^2(x)} - \frac{1}{sin(x)}$

$\frac{cos(x)}{sin^2(x)} - \frac{sin(x)}{sin^2(x)} = \frac{cos(x) - sin(x)}{sin^2(x)}$

We can't simplify it more.

Second case:

If the initial equation was

$\frac{cot(x)}{sin(x) - csc(x)}$

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

$\frac{cos(x)}{sin(x)}*\frac{1}{sin(x) - 1/sin(x)} = \frac{cos(x)}{sin(x)}*\frac{1}{sin^2(x)/sin(x) - 1/sin(x)}$

This is equal to:

$\frac{cos(x)}{sin(x)}*\frac{sin(x)}{sin^2(x) - 1}$

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:

$\frac{cos(x)}{sin(x)}*\frac{sin(x)}{sin^2(x) - 1} = \frac{cos(x)}{sin(x)}*\frac{sin(x)}{-cos^2(x)} = -\frac{cos(x)}{cos^2(x)}*\frac{sin(x)}{sin(x)} = - \frac{1}{cos(x)}$