Choose the right expression in terms of a single trigonometric function. cos space x space cos space 2 x space plus space sin space x space sin space 2 x
Are you saying
cosx cos(2x) + sinx sin(2x) ??? , then why not type it that way?
You should recognize the formula
cos(A - B) = cosAcosB + sinAsinB
cosx cos(2x) + sinx sin(2x)
= cos(x - 2x)
= cos(-x)
= cosx
To simplify the expression cos(x)cos(2x) + sin(x)sin(2x) in terms of a single trigonometric function, you can use the trigonometric identity called the angle sum identity.
The angle sum identity for cosine states that cos(A + B) = cos(A)cos(B) - sin(A)sin(B).
In this case, let's rewrite the expression as follows:
cos(x)cos(2x) + sin(x)sin(2x)
By applying the angle sum identity for cosine, we can see that the expression resembles cos(A + B). Let's represent A = x and B = 2x:
cos(A + B) = cos(A)cos(B) - sin(A)sin(B)
Now, substitute A = x and B = 2x into the identity:
cos(x + 2x) = cos(x)cos(2x) - sin(x)sin(2x)
Combining like terms, we have:
cos(3x) = cos(x)cos(2x) - sin(x)sin(2x)
Therefore, the expression cos(x)cos(2x) + sin(x)sin(2x) can be simplified to cos(3x).