De Moivre's Theorem

The Chebyshev polynomials of the first kind are a class of polynomials denoted $T_n$ for any nonnegative integer $n.$ (The T is for Tchebyshev, the French way of spelling Chebyshev.) There are a variety of ways to define these polynomials (like with differential equations), but the most famous is: $\\$ $$T_0(x) = 1\\ T_1(x) = x\\ T_{n+1}(x) = 2xT_n(x) - T_{n-1}(x)$$ In addition, $$T_n(\cos(x))=\cos(nx)$$ $\\$ For example, $\cos(2x)=2\cos^2(x)-1$, so $T_2 = 2x^2-1$. $$\\$$ Using this information, find $T_4 (x).$ $\\$ (Hint: use De Moivre's formula to expand $\cos(4x)+i \sin(4x)$, then compare the real and imaginary parts).