Perform the multiplication and use the fundamental identities to simplify.
(cotx + cscx)(cotx-cscx)
I know that you have to foil first so
cot^2x - csc^2x and then use the pythagorean identity 1+cot^2u = csc^2u but I do not know how to simplify.
To simplify the expression (cotx + cscx)(cotx - cscx), you correctly FOIL (First, Outer, Inner, Last) the terms:
(cotx + cscx)(cotx - cscx)
= cotx * cotx - cotx * cscx + cscx * cotx - cscx * cscx
Now, to simplify further, we can use the reciprocal identities:
1. cotx = cosx/sinx
2. cscx = 1/sinx
Substituting these values into the expression:
(cotx + cscx)(cotx - cscx)
= (cosx/sinx)(cosx/sinx) - (cosx/sinx)(1/sinx) + (1/sinx)(cosx/sinx) - (1/sinx)(1/sinx)
Now, simplify each term:
= (cosx * cosx) / (sinx * sinx) - (cosx * 1) / (sinx * sinx) + (1 * cosx) / (sinx * sinx) - (1 * 1) / (sinx * sinx)
= cos^2x / sin^2x - cosx / sin^2x + cosx / sin^2x - 1 / sin^2x
Now, combine the terms with the same denominator:
= (cos^2x - cosx + cosx - 1) / sin^2x
= (cos^2x - 1) / sin^2x
Using the Pythagorean identity sin^2x + cos^2x = 1, we can simplify further:
= (1 - 1) / sin^2x
= 0 / sin^2x
= 0
Therefore, (cotx + cscx)(cotx - cscx) simplifies to 0.