transform the expression on left to one on right

Question

csc x-sin x to cot x cos x

Answer 1

csc x-sin x = (1/sinx) - sin x
= (1 - sin^2 x)/sin x = (cos x * cos x)/sin x
= cos x cot x

Answer 2

We have trig identities.

cscx - sinx to cotx(cos x).

cscx is the reciprocal of the sine function.

So, then:

cscx = 1/sinx

We now have:

1/sinx - sinx...Treat this like a fraction case.

1/sinx - sinx becomes (1 - sin^2x)/sinx

Of course, (1 - sin^2x) is one of the Pythagorean Identities, recall?

We know that (1 - sin^2x) = cos^2x

We now have:

cos^2x/sinx = the right side

Done!