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).