Questions: Factor and simplify the expression sin^3 x + sin x cos^2 x into one term involving one trigonometric function.

Solution Steps

To factor and simplify the expression \(\sin^3 x + \sin x \cos^2 x\), we can start by factoring out the common term \(\sin x\). This will allow us to express the entire expression in terms of a single trigonometric function. After factoring, we can use trigonometric identities to further simplify the expression.

We start with the expression

\[ \sin^3 x + \sin x \cos^2 x. \]

We can factor out the common term \(\sin x\):

\[ \sin x (\sin^2 x + \cos^2 x). \]

Next, we apply the Pythagorean identity, which states that

\[ \sin^2 x + \cos^2 x = 1. \]

Substituting this identity into our factored expression gives us:

\[ \sin x (1). \]

Thus, the expression simplifies to:

\[ \sin x. \]

Final Answer