combine fractions and simplify to a multiple of a power of a basic trig. function.?

sin x/cot²x - sin x/cos²x

To combine and simplify these fractions, we can first find a common denominator. The denominator of the first fraction is cot²x, and the denominator of the second fraction is cos²x.

To find a common denominator, we need to identify the least common multiple (LCM) of cot²x and cos²x. Since cot²x is equivalent to 1/tan²x, we can rewrite it as (1/tan²x)(cos²x/cos²x), which simplifies to cos²x/(tan²x * cos²x). Now we can see that the LCM of cot²x and cos²x is (tan²x * cos²x).

Next, we can adjust the numerators of both fractions to have the same denominator:

sin x/cot²x = (sin x)(cos²x)/(tan²x * cos²x)

sin x/cos²x = (sin x * tan²x * cos²x)/(tan²x * cos²x)

Now, we can combine the fractions:

(sin x)(cos²x)/(tan²x * cos²x) - (sin x * tan²x * cos²x)/(tan²x * cos²x)

To simplify further, we can cancel out the common factors:

(sin x)(cos²x) - (sin x * tan²x * cos²x)/(tan²x * cos²x)

This simplifies to:

(sin x * cos²x - sin x * tan²x * cos²x)/(tan²x * cos²x)

Factoring out sin x, we have:

sin x * (cos²x - tan²x * cos²x)/(tan²x * cos²x)

Simplifying the expression in the parentheses:

cos²x - tan²x * cos²x = cos²x(1 - tan²x)

We know that 1 - tan²x = sec²x, so we can substitute that:

cos²x(1 - tan²x) = cos²x(sec²x)

Putting it all together, we have:

sin x * cos²x(sec²x)/(tan²x * cos²x)

The cos²x terms in the numerator and denominator cancel out:

sin x * sec²x/tan²x

This can be further simplified using the identity sec x = 1/cos x and tan x = sin x/cos x:

(sin x/cos x) * (1/cos²x) / (sin²x/cos²x)

Canceling out the common factors:

(1/cos x) * (1/cos²x) / (sin²x/cos²x)

Multiplying the fractions:

1/(cos x * cos²x) / (sin²x/cos²x)

Inverting the denominator:

1/(cos x * cos²x) * (cos²x/sin²x)

Canceling out the common factor cos²x:

1/(cos x * cos²x) * (1/sin²x)

Simplifying:

1/(cos x * cos²x * sin²x)

Now, we have the combined and simplified expression: 1/(cos x * cos²x * sin²x).