Evaluate the identity
(cos^2x-sin^2x)/(1-tan^2x)=cosx^2X
To evaluate the given identity, we need to simplify the left-hand side (LHS) and show that it is equal to cos^2x.
Let's simplify the LHS step by step:
1. Start with the given identity:
(cos^2x - sin^2x) / (1 - tan^2x) = cos^2x
2. Rewrite sin^2x and tan^2x in terms of cos^2x:
(cos^2x - (1 - cos^2x)) / (1 - (sin^2x / cos^2x))
Simplifying (1 - cos^2x) yields sin^2x:
(2cos^2x - 1) / (cos^2x - sin^2x / cos^2x)
3. Simplify the denominator:
cos^2x - (sin^2x / cos^2x) can be simplified to:
cos^2x - (1 / (cos^2x / sin^2x))
Using the identity sin^2x + cos^2x = 1, we can rewrite it as:
cos^2x - (1 / (1 / cos^2x))
cos^2x - cos^2x
4. Simplify the numerator:
(2cos^2x - 1) - cos^2x
Distribute the negative sign:
2cos^2x - 1 - cos^2x
Combine like terms:
cos^2x - 1
5. Simplify the LHS further:
(cos^2x - 1) / (cos^2x - cos^2x)
cos^2x - 1 / 0
At this point, we have hit a contradictory statement, which means that the given identity is not valid.
Hence, the evaluation of the identity (cos^2x - sin^2x) / (1 - tan^2x) to cos^2x is incorrect.