Having trouble with this problem:
Rewrite the expression as an equivalent expression that does not contain powers greater than 1.
Cos³x
My answer: cos³x=(cos²x)(cosx)
=(1+cos2x/2)(cosx)-->cos+cos³x/2...which is wrong
You have to use doubling formulas again to express it in terms of cos(3x). If you know about complex numbers then the derivation is easy:
cos(x) = [exp(i x) + exp(-i x)]/2
cos^3(x) = 1/8 [exp(3 i x) +
exp(-3 i x)+ 3 exp(ix) + 3 exp(-ix) ] =
1/4 cos(3 x) + 3/4 cos(x)
(cosx)+(cosx)(cosx)
To rewrite the expression cos³x as an equivalent expression that does not contain powers greater than 1, you can use the identity:
cos²x = 1 - sin²x
We'll substitute this identity into the expression:
cos³x = (cos²x)(cosx)
cos³x = (1 - sin²x)(cosx)
So the equivalent expression that does not contain powers greater than 1 is (1 - sin²x)(cosx).
To rewrite the expression cos³x as an equivalent expression without powers greater than 1, you can use the trigonometric identity for the cube of cosine:
cos³x = (cosx)³ = cosx * cosx * cosx
Since the question asks for an expression without powers greater than 1, you can rewrite cos³x as:
cos³x = cosx * cos²x
However, it seems like you want to simplify the expression further. Let's continue with that:
Using another trigonometric identity, we have:
cos²x = 1 - sin²x
Substituting this into the expression, we get:
cos³x = cosx * (1 - sin²x)
Now, expanding the expression:
cos³x = cosx - cosx * sin²x
This is an equivalent expression to cos³x that does not contain powers greater than 1.