How do you simplify sqrt((1-costheta)(1+costheta))?

2 Answers

It is

sqrt((1-costheta)(1+costheta))=sqrt[1-cos^2 theta]= sqrt[sin^2 theta]=abs(sin theta)

where abs is the absolute value.

We used the fact that sin^2 theta+cos^2 theta=1

Jul 23, 2016

sqrt((1-costheta)(1+costheta)) = |sintheta|

Explanation:

For this problem, we can use the difference of squares methods, which tells us that

(a-b)(a+b) = a^2-b^2

Applying this method, we then get

sqrt((1-costheta)(1+costheta)) = sqrt(1-cos^2theta)

Also note that sin^2theta + cos^2theta = 1 -> sin^2theta = 1-cos^2theta

Thus we can simplify this even further, giving us

sqrt(1-cos^2theta) = sqrt(sin^2theta)

We have to be careful here, because the root implies that we should have to answers. In fact, if we would have made the mistake of saying that sqrt(sin^2theta) = sin x, we would have the following graphs:

Graph of sqrt(sin^2theta):
graph{sqrt(sin x * sinx) [-5.504, 5.596, -2.153, 3.396]}
Graph of sin theta:
graph{sin x [-5.504, 5.596, -2.153, 3.396]}

These functions are noticeably different, but they're only off by a little bit. In this problem, we could say that

sqrt((1-costheta)(1+costheta))

= sqrt(sin^2theta) = -sintheta and sin theta -> |sintheta|