Question

How do you simplify `sin theta cos ^2 theta - sin theta`?

Answer 1

`-sin^2(theta)`

Explanation:

Remember that
`color(white)("XXX")sin^2(theta)+cos^2(theta)=1`
or (in a modified form)
`color(white)("XXX")color(blue)(cos^2(theta)-1=-sin^2(theta))`

Given the expression:
`color(white)("XXX")sin(theta)cos^2(theta)-sin(theta)`
we can factor this as
`color(white)("XXX")color(red)(sin(theta))*(color(blue)(cos^2(theta)-1))`

The using the earlier identity, we have
`color(white)("XXX")color(red)(sin(theta))*(color(blue)(-sin(theta)))`
or
`color(white)("XXX")-sin^2(theta)`