Problem

This problem asks you to express $\cos 5\theta$ as a polynomial in $\cos \theta$—a classic application of Chebyshev polynomials. The key insight is to use the angle-addition formula or the recurrence relation $\cos(n\theta)$ repeatedly: each time you add another $\theta$, you can rewrite the result using the identity $\cos^2 \theta = 1 - \sin^2 \theta$ to eliminate higher powers of sine. By systematically building up from $\cos \theta$ to $\cos 5\theta$, you'll discover that the coefficients $a, b, c, d$ follow a predictable pattern tied to powers of $\cos \theta$.

This problem asks you to express $\cos 5\theta$ as a polynomial in $\cos \theta$—a classic application of Chebyshev polynomials. The key insight is to use the angle-addition formula or the recurrence relation $\cos(n\theta)$ repeatedly: each time you add another $\theta$, you can rewrite the result using the identity $\cos^2 \theta = 1 - \sin^2 \theta$ to eliminate higher powers of sine. By systematically building up from $\cos \theta$ to $\cos 5\theta$, you'll discover that the coefficients $a, b, c, d$ follow a predictable pattern tied to powers of $\cos \theta$.