Prove the identity: cosx/1-tanx +sinx/1-cotx
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To prove the given identity, we will first simplify each term individually, then combine them.
1. Simplifying cos(x) / (1 - tan(x)):
We know that tan(x) = sin(x) / cos(x). Substitute this value into the denominator:
1 - tan(x) = 1 - sin(x) / cos(x)
To combine the fractions in the denominator, we need a common denominator. The common denominator is cos(x), so multiply the first term by cos(x)/cos(x):
cos(x) / (1 - tan(x)) = cos(x) / (1 - sin(x) / cos(x))
Multiply the numerator and denominator by cos(x) to simplify further:
cos(x) / (1 - tan(x)) = cos^2(x) / (cos(x) - sin(x))
2. Simplifying sin(x) / (1 - cot(x)):
We know that cot(x) = cos(x) / sin(x). Substitute this value into the denominator:
1 - cot(x) = 1 - cos(x) / sin(x)
To combine the fractions in the denominator, we need a common denominator. The common denominator is sin(x), so multiply the first term by sin(x)/sin(x):
sin(x) / (1 - cot(x)) = sin(x) / (1 - cos(x) / sin(x))
Multiply the numerator and denominator by sin(x) to simplify further:
sin(x) / (1 - cot(x)) = sin^2(x) / (sin(x) - cos(x))
3. Combine the simplified fractions:
cos^2(x) / (cos(x) - sin(x)) + sin^2(x) / (sin(x) - cos(x))
Since the denominators are now the same, we can combine the numerators:
(cos^2(x) + sin^2(x)) / (cos(x) - sin(x))
By applying the identity cos^2(x) + sin^2(x) = 1, we get:
1 / (cos(x) - sin(x))
This matches the right-hand side of the given identity, so we have proven the given identity:
cos(x) / (1 - tan(x)) + sin(x) / (1 - cot(x)) = 1 / (cos(x) - sin(x))