simplify (1-cot^2x sec^2x) / ( cot^2x)
To simplify the expression (1 - cot^2x sec^2x) / cot^2x, we can use some trigonometric identities.
Step 1: Simplify the numerator:
Recall that the Pythagorean identity is sec^2x = 1 + tan^2x, so we can rewrite sec^2x as (1 + tan^2x).
So, the numerator becomes:
(1 - cot^2x (1 + tan^2x))
Step 2: Simplify the expression within the parentheses:
Using the reciprocal identity, cot^2x = 1/tan^2x, we can substitute it into the expression within the parentheses.
The expression becomes:
(1 - (1/tan^2x)(1 + tan^2x))
Simplifying further, we get:
(1 - 1) = 0
Step 3: Simplify the denominator:
Since the denominator is cot^2x, we can express it as 1/tan^2x using the reciprocal identity.
The expression becomes:
(0) / (1/tan^2x)
Simplifying further, we multiply by the reciprocal of the denominator, which is tan^2x/1:
0 * (tan^2x/1) = 0
Therefore, the simplified form of the expression (1 - cot^2x sec^2x) / cot^2x is simply 0.