Question

Sydney simplifies the expression 2csc(-y) with the work below. What did she do wrong? 2csc(x+y)= 2/sin(x-y)

Answer 1

Answer:

We know that:

csc(x) = 1/sin(x)

Now we want to work with:

2csc(x+y)

The first step is:

2csc(x+y)= 2/sin(x-y)

This is correct so far.

Now we need to use the relation:

sin(a + b) = sin(a)*cos(b) + sin(b)*cos(a)

Then:

sin(x - y) = sin(x)*cos(-y) + sin(-y)*cos(x)

sin(x) is an odd function, then: sin(-y) = -sin(y)

and cos(x) is an even function, then:

cos(-y) = cos(y)

Then we get:

sin(x - y) = sin(x)*cos(y) - sin(y)*cos(x)

then:

2/sin(x-y) = 2/(sin(x)*cos(y) - sin(y)*cos(x))

Now let's look at what she did:

2/sin(x-y) = 2/(cos x cos(-y)+sin xsin(-y))

Here is her error, she replaced:

sin(x - y) by cos x cos(-y)+sin(x)sin(-y)

This relation is for the cosine function, not the sine one, so here is her mistake.