Rút gọn sin^2x-tan^2x/cos^2x-cot^2x

To simplify the expression sin^2x_tan^2x/cos^2x-cot^2x, we can start by simplifying the individual terms.

First, let's simplify sin^2x_tan^2x. We know that tan^2x = sin^2x/cos^2x, so we can substitute this into the expression:

sin^2x_tan^2x = sin^2x * (sin^2x/cos^2x) = sin^4x/cos^2x

Next, let's simplify cos^2x-cot^2x. We know that cot^2x = cos^2x/sin^2x, so we can substitute this into the expression:

cos^2x-cot^2x = cos^2x - (cos^2x/sin^2x) = cos^2x - cos^2x/sin^2x

To simplify further, we can find a common denominator for the two terms:

cos^2x - cos^2x/sin^2x = (cos^2x * sin^2x - cos^2x) / sin^2x

Now, let's simplify the numerator:

(cos^2x * sin^2x - cos^2x) = cos^2x * (sin^2x - 1)

Finally, we can substitute the simplified terms back into the original expression:

sin^4x/cos^2x - (cos^2x * (sin^2x - 1)) / sin^2x

Now, we can simplify further by finding a common denominator for the two terms:

(sin^4x * sin^2x - cos^2x * (sin^2x - 1)) / (cos^2x * sin^2x)

Next, let's simplify the numerator:

(sin^4x * sin^2x - cos^2x * (sin^2x - 1)) = sin^6x - cos^2x * sin^2x + cos^2x

Finally, we can substitute the simplified terms back into the original expression:

(sin^6x - cos^2x * sin^2x + cos^2x) / (cos^2x * sin^2x)

And that is the simplified expression.