Clenshaw-Curtis quadrature is based on writing $$ \int_{-1}^{1} f(x)dx=\int_{0}^{\pi}f(\cos y)\sin y dy $$ and then replacing $f(\cos y)$ by a truncated Fourier series, so that the integral can be written as sum over these Fourier coefficients.
Why is it necessary to compute that Fourier series? Why can one not simply apply the trapezoidal rule to $f(\cos y)\sin y$? As far as I understand this should yield a quadrature rule with the same convergence rates (due to Euler-McLaurin) and same quadrature nodes but easier (numerically and conceptually) quadrature weights than Clenshaw-Curtis. Am I missing something?
Clenshaw-Curtis is interpolatory but the trapezoidal rule for the transformed function is not. Thus for $f(x)=1$ Clenshaw-Curtis will always spit out $$\int_{-1}^1f(x)dx\approx2$$ But trapezoidal rule applied to the transformed function will result in $$\begin{align}\int_{-1}^1f(x)dx&=\int_0^{\pi}1\cdot\sin\theta d\theta\\ &\approx\frac{\pi}{2N}\left[\sin(0)+2\sum_{k=1}^{N-1}\sin\frac{k\pi}N+\sin(\pi)\right]\\ &=\frac{\pi}N\sum_{k=1}^{N-1}\frac{-1}{2\sin\frac{\pi}{2N}}\left(\cos\frac{\left(k+\frac12\right)\pi}N-\cos\frac{\left(k-\frac12\right)\pi}N\right)\\ &=\frac{-\pi}{2N\sin\frac{\pi}{2N}}\left(\cos\frac{\left(N-\frac12\right)\pi}N-\cos\frac{\pi}{2N}\right)\\ &=\frac{\pi}N\cot\frac{\pi}{2N}\approx2\left(1-\frac{\pi^2}{12N^2}\right)\end{align}$$ For a large number of subintervals $N$. Thus Clenshaw-Curtis is generally considered to use the functional values at the same points as the proposed method more efficiently.