For a continuously differentiable function $f$ on $[-\pi, \pi]$, prove that $$\Bigg|\int_{-\pi}^\pi \Big(f(x) \cos(x) - f'(x)\sin(x)\Big)\,dx\Bigg| \leq \sqrt{2\pi}\Bigg\{\int_{-\pi}^\pi \Big(|f(x)|^2 + |f'(x)|^2\Big) \, dx\Bigg\}^{1/2}$$
My try: $$ \left| \int_{-\pi}^\pi \Big(f(x) \cos(x) - f'(x)\sin(x)\Big)\,dx \right| \leq \int_{-\pi}^\pi \Big|f(x) \cos(x) - f'(x)\sin(x)\Big| \, dx $$
Now, I am supposed to use some Inner Product on $C^1[a,b]$ here but I don't know which one to use and how. It would be great to have a hint. Thanks.