sin x / 1-cosx is equivalent to:

sin x / 1 + cos x Is this right?

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No. Since the numerators are equal, the denominators would have to be equal.

It might also be of interest to note that:
sin x/(1-cosx) = cot(x/2)
sin x/(1+cosx) = tan(x/2)

The product of the two is equal to 1, but they are not equal to each other.

It IS true that
sin x/(1-cosx) = (1+cosx)/sinx

To determine if sin x / (1 - cos x) is equivalent to sin x / (1 + cos x), we can simplify both expressions and compare them.

Starting with sin x / (1 - cos x):

Step 1: Multiply the numerator and denominator by the conjugate of the denominator, which is (1 + cos x). This helps us simplify the expression.

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

This gives us:

(sin x * (1 + cos x)) / (1 - cos^2 x)
Since cos^2 x = (cos x) * (cos x), we can substitute it in the denominator:

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

Simplifying, we have:

(sin x + sin x * cos x) / (1 - cos^2 x)
Using the identity sin^2 x + cos^2 x = 1, we can substitute 1 - cos^2 x with sin^2 x:

(sin x + sin x * cos x) / sin^2 x
Factoring out sin x in the numerator:

sin x * (1 + cos x) / sin^2 x

Now, let's simplify sin x / (1 + cos x):

Sin x / (1 + cos x)

This expression already appears to be in its simplest form.

Comparing the two expressions, we have:

(sin x * (1 + cos x)) / (sin^2 x) and sin x / (1 + cos x)

After simplifying both expressions, we see that they are NOT equivalent.