Show algebraically how to confirm that cos2x=cos^2x-sin^2x using the sum and difference identities
no
To confirm that cos(2x) = cos^2(x) - sin^2(x) using the sum and difference identities, we can start by using the double angle identity for cosine, which states:
cos(2x) = cos^2(x) - sin^2(x)
Now let's express cos(2x) using the sum and difference identities. We know that cos(A + B) = cos(A)cos(B) - sin(A)sin(B).
In this case, we can rewrite 2x as the sum of two angles: A = x and B = x. Therefore, we can rewrite cos(2x) as:
cos(2x) = cos(x + x)
Using the sum identity, we have:
cos(2x) = cos(x)cos(x) - sin(x)sin(x)
Notice that cos(x)cos(x) is equal to cos^2(x), and sin(x)sin(x) is equal to sin^2(x).
Therefore, we can simplify the equation to:
cos(2x) = cos^2(x) - sin^2(x)
Thus, we have algebraically confirmed that cos(2x) = cos^2(x) - sin^2(x) using the sum and difference identities.
huh?
cos 2 x = cos ( x+x )
= cos x cos x - sin x sin x
= cos^2 x - sin^2 x