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.