Verify the identity cscx+cotx=sinx/(1-cosx).

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giorgiana1976 | College Teacher | (Level 3) Valedictorian

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We'lll manage only the left side and we'll re-write the terms of the sum:

csc x = 1/sin x and cot x = cos x/sin x

csc x + cot x = 1/sin x + cos x/sin x

Since the fractions have the same denominator, we can write:

csc x + cot x = (1+cos x)/sin x

The expression will become:

(1+cos x)/sin x = sin x/(1 - cos x)

We'll cross multiply:

(sin x)^2 = (1+cos x)(1-cos x)

The product from the right side returns the difference of two squares:

(sin x)^2 = 1 - (cos x)^2

We'll add (cos x)^2 both sides:

(sin x)^2 + (cos x)^2 = 1

But Pythagorean identity states that (sin x)^2 + (cos x)^2 = 1, therefore we've came up with a true statement.

Therefore, the given identity csc x + cot x = sin x/(1 - cos x) is verified.