Simplify sin x cos^2x-sinx
Here's my book's explanation which I don't totally follow
sin x cos^2x-sinx=sinx(cos^2x-1)
=-sinx(1-cos^2x)
=-sinx(sin^2x) (Where does sine come from and what happend to cosine?)
=-sin^3x
To understand the simplification of the expression sin x cos^2x - sin x, let's break it down step by step:
1. Start with the given expression: sin x cos^2x - sin x.
2. The first step is to factor out the common term, sin x:
sin x (cos^2x - 1).
3. Now we have the expression sin x multiplied by the quantity (cos^2x - 1). To simplify it further, we can use the identity for cosine squared, which states that cos^2x = 1 - sin^2x. Replacing cos^2x with 1 - sin^2x, we get:
sin x (1 - sin^2x - 1).
4. Simplify the expression by combining like terms within the parentheses:
sin x (-sin^2x).
5. Finally, we can rearrange the expression by multiplying the terms together:
-sin^3x.
So, the simplified form of sin x cos^2x - sin x is -sin^3x.
sin x cos^2x-sinx=sinx(cos^2x-1) they took out a common factor of sinx
=-sinx(1-cos^2x) recall that sin^2x + cos^2x = 1, and then 1-cos^2x = sin^2x.
Notice they had cos^2x-1 which is -(1-cos^2x). Also notice that there is now a - in front of the sinx
=-sinx(sin^2x) (Where does sine come from and what happend to cosine?)
=-sin^3x
does it make sense now?