Which expression is equal to csc x, written in termsof cos x ?

Two trig identities are of interest here:

... sin(x)² + cos(x)² = 1

... csc(x) = 1/sin(x)

Using the first one, you can subtract cos(x)², the take the square root to find

... sin(x) = ±√(1 - cos(x)²)

Using that in the second one, you have

... csc(x) = ±1/√(1 - cos(x)²) . . . . . . . . . . . . . matches the 3rd choice

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this is a good question! i'll do some research and come back when i have the answer!