simplify the expression.

cos^2x+sin^2x/cot^2x-csc^2x

Parentheses are assumed missing. Implied parentheses are ALWAYS required in numerator and denominator of fractions.

(cos^2x+sin^2x)/(cot^2x-csc^2x)

This problem can be solved by converting all functions in terms of sine and cosine according to the standard definitions.

(cos^2x+sin^2x)/(cot^2x-csc^2x)
=(cos^2(x)+sin^2(x))/(cos^2(x)/sin^2(x)-1/(sin^2(x))

Use sin²(u)+cos²(u)=1 to reduce the numerator to 1.
Since the denominator has a common factor of sin²(x), we can simplify that too!

=(1)/[(cos²(x)-1)/sin²(x)]
=sin²(x)/(cos²(x)-1)
=sin²(x)/(-sin²(x)
=-1

To simplify the expression (cos^2x + sin^2x) / (cot^2x - csc^2x), we can start by simplifying the numerator and denominator separately.

Numerator: cos^2x + sin^2x
Using the Pythagorean identity sin^2x + cos^2x = 1, we can simplify the numerator to 1.

Denominator: cot^2x - csc^2x
Using the reciprocal identities cotx = 1/tanx and cscx = 1/sinx, we can rewrite the denominator as (1/tan^2x) - (1/sin^2x).
Next, we can find the common denominator by multiplying the first fraction by sin^2x/sin^2x:
(1/tan^2x) - (1/sin^2x) = (sin^2x/(tan^2x*sin^2x)) - (tan^2x/(tan^2x*sin^2x)) = (sin^2x - tan^2x)/(tan^2x*sin^2x).

Simplifying the denominator further:
Using the Pythagorean identity tan^2x + 1 = sec^2x, we can rewrite the denominator as ((1 - tan^2x)/tan^2x) / (tan^2x*sin^2x).
Simplifying the numerator: (1 - tan^2x) can be rewritten as -tan^2x.

Now, we have -tan^2x / (tan^2x*sin^2x).
Since the negative (-) sign applies to both the numerator and denominator, we can cancel them out, resulting in:
tan^2x / (tan^2x*sin^2x).

Finally, we can cancel out the common factor of tan^2x, resulting in:
1/sin^2x.

Therefore, the expression (cos^2x + sin^2x) / (cot^2x - csc^2x) simplifies to 1/sin^2x.

To simplify the expression cos^2x + sin^2x / cot^2x - csc^2x, we will first simplify the numerator and denominator separately, and then simplify the resulting expression.

Numerator: cos^2x + sin^2x
We know that cos^2x + sin^2x is equal to 1, according to the Pythagorean identity for trigonometric functions.

Denominator: cot^2x - csc^2x
To simplify the denominator, we need to express cot^2x and csc^2x in terms of sine and cosine.

cot^2x can be expressed as 1 / tan^2x. Since tan^2x = sin^2x / cos^2x, we have cot^2x = 1 / (sin^2x / cos^2x) = cos^2x / sin^2x.

csc^2x can be expressed as 1 / sin^2x.

Therefore, the denominator becomes: cos^2x / sin^2x - 1 / sin^2x.

Combining the terms in the denominator, we get: (cos^2x - 1) / sin^2x.

Now that we have simplified the numerator and denominator separately, we can simplify the entire expression.

cos^2x + sin^2x / cot^2x - csc^2x = 1 / [(cos^2x - 1) / sin^2x]
Applying the reciprocal of the denominator by multiplying numerator and denominator by sin^2x, we get:
= 1 * sin^2x / (cos^2x - 1)
= sin^2x / (cos^2x - 1)

Therefore, the simplified expression is sin^2x / (cos^2x - 1).