Final answer:
To simplify the expression cos(x)tan(x) - sin(x)cos^2(x), use trigonometric identities to rewrite the expression and then simplify it using the identity sin^2(x) + cos^2(x) = 1. The simplified expression is sin^3(x), so the correct answer is (D) sin^2(x).
Step-by-step explanation:
To simplify the expression cos(x)tan(x) - sin(x)cos^2(x), we can use trigonometric identities to rewrite the expression. First, recall that tan(x) = sin(x)/cos(x). Substituting this into the expression, we have:
cos(x) * (sin(x)/cos(x)) - sin(x)cos^2(x)
Simplifying further, we get:
sin(x) - sin(x)cos^2(x)
Factoring out sin(x) gives us:
sin(x)(1 - cos^2(x))
Using the identity sin^2(x) + cos^2(x) = 1, we can simplify the expression as:
sin(x)(sin^2(x))
The simplified expression is sin^3(x), so the correct answer is (D) sin^2(x).