Final answer:
To show that there is a polynomial P(t) of degree 4 such that cos 4x = P(cos x), we can expand cos 4x using the angle-sum identity and express it in terms of cos x.
Step-by-step explanation:
To show that there is a polynomial P(t) of degree 4 such that cos 4x = P(cos x), we can expand cos 4x using the angle-sum identity:
cos 4x = cos(2x + 2x) = cos²(2x) - sin²(2x)
Now, using the double-angle identities, we can express cos²(2x) and sin²(2x) in terms of cos x and sin x:
cos²(2x) = (1 + cos(2x))/2 = (1 + 2cos²(x) - 1)/2 = cos²(x)
sin²(2x) = (1 - cos(2x))/2 = (1 - 2cos²(x) + 1)/2 = 1 - cos²(x)
Substituting these values back into the previous expression:
cos 4x = cos²(x) - (1 - cos²(x)) = 2cos²(x) - 1
Therefore, the polynomial P(t) = 2t² - 1 satisfies cos 4x = P(cos x).