Answer:
(cot x + 6) / cos x Or (sin x + 6)/cos x
Explanation:
We start with the expression:
(cot^2 x + 12 cot x +36)/ (cos x *cot x + 6 cos x)
We can factor the numerator:
(cot^2 x + 12 cot x +36) = (cot x + 6)^2
Now we can rewrite the denominator as:
cos x * cot x + 6 cos x = cos x * (cot x + 6)
We can substitute these expressions into the original expression:
(cot^2 x + 12 cot x +36)/ (cos x *cot x + 6 cos x) = (cot x + 6)^2 / (cos x * (cot x + 6))
We can simplify further by canceling out the common factor of
(cot x + 6):
(cot x + 6)^2 / (cos x * (cot x + 6)) = (cot x + 6) / cos x
Therefore, the simplified expression with no fractions is:
(cot^2 x + 12 cot x +36)/ (cos x *cot x + 6 cos x) = (cot x + 6) / cos x
Alternatively, we can write the simplified expression in a more expanded form:
(cot^2 x + 12 cot x +36)/ (cos x *cot x + 6 cos x) = (cot x/cos x + 6/cos x)
Using the identity cot x = cos x/sin x, we can rewrite the first term:
cot x/cos x = cos x/sin x * 1/cos x = sin x/cos x
Now we have:
(cot^2 x + 12 cot x +36)/ (cos x *cot x + 6 cos x) = (sin x/cos x + 6/cos x)
Combining the two terms, we get:
(sin x/cos x + 6/cos x) = (sin x + 6)/cos x
Therefore, another possible simplified expression with no fractions is:
(cot^2 x + 12 cot x +36)/ (cos x *cot x + 6 cos x) = (sin x + 6)/cos x