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.
