Suppose f(x)=cos x - sin x and g(x)=cos x + sin x. Explain why the graph of (fxg)(x) is equivalent to the graph of h(x)=cos x after it has been horizontally compressed by a factor of 1/2. Thanks so much...
To demonstrate why the graph of (f * g)(x) is equivalent to the graph of h(x) = cos(x) after being horizontally compressed by a factor of 1/2, we need to show that (f * g)(x) = h(2x).
First, let's find the expression for (f * g)(x):
(f * g)(x) = f(x) * g(x)
= (cos(x) - sin(x)) * (cos(x) + sin(x))
= cos^2(x) - sin^2(x)
= cos^2(x) - (1 - cos^2(x)) [Using the identity sin^2(x) = 1 - cos^2(x)]
= 2cos^2(x) - 1
Now, let's substitute 2x into h(x) and examine its form:
h(2x) = cos(2x)
Using the double angle formula cosine identity, cos(2x) can be expressed as:
cos(2x) = 2cos^2(x) - 1
Comparing the expression for (f * g)(x) and h(2x), we can see that they are identical:
(f * g)(x) = h(2x)
This shows that the graph of (f * g)(x) is equivalent to the graph of h(x) = cos(x) after it has been horizontally compressed by a factor of 1/2.