Final answer:
Yes, there is a polynomial Q(t) of degree 5 such that cos(5x) = Q(cos(x)); this can be shown using trigonometric identities and multiple-angle formulae for cosine, expressing cos(5x) as a linear combination of powers of cos(x).
Step-by-step explanation:
Show that there is a polynomial of degree 5 such that cos(5x) = Q(cos(x))
To show that there exists a polynomial Q(t) of degree 5 such that cos(5x) = Q(cos(x)), we can utilize the multiple-angle identities from trigonometry. The multiple-angle identity for cosine, specifically for cos(5x), involves powers of cos(x). This identity can be written as a linear combination of different powers of cos(x) up to the fifth power.
Starting from the cosine multiple-angle formulae, we expand cos(5x) into a series involving powers of cos(x). The expansion is derived using trigonometric identities and involves binomial coefficients. This expansion will yield a polynomial expression - a sum of terms where each term is a constant coefficient multiplied by a power of cos(x). Since we are considering cos(5x), it's guaranteed that the highest power will be five due to the nature of the trigonometric expansion.
Therefore, after some algebraic manipulations, we see that it is indeed possible to express cos(5x) as a polynomial Q(t) of cos(x) where t = cos(x) and the highest power in that polynomial is 5, validating the student's inquiry.