Prove this inequality with Cauchy-Schwarz inequality

let $x_{1},x_{2},\cdots,x_{n}>0$, show that

$$\left(\sum_{k=1}^{n}x_{k}\cos{k}\right)^2+\left(\sum_{k=1}^{n}x_{k}\sin{k}\right)^2\le \left(2+\dfrac{n}{4}\right)\sum_{k=1}^{n}x^2_{k}$$

I can prove if $2+\dfrac{n}{4}$ take the place of $n$,

it seem can use Cauchy-Schwarz inequality $$\left(\sum_{k=1}^{n}x_{k}\cos{k}\right)^2\le\sum_{k=1}^{n}x^2_{k}\sum_{k=1}^{n}\cos^2{k}\tag{1}$$

$$\left(\sum_{k=1}^{n}x_{k}\sin{k}\right)^2\le\sum_{k=1}^{n}x^2_{k}\sum_{k=1}^{n}\sin^2{k}\tag{2}$$ since $(1)+(2)$ we have $$\left(\sum_{k=1}^{n}x_{k}\cos{k}\right)^2+\left(\sum_{k=1}^{n}x_{k}\sin{k}\right)^2\le\sum_{k=1}^{n}x^2_{k}\sum_{k=1}^{n}(\cos^2{k}+\sin^2{k})=n\sum_{k=1}^{n}x^2_{k}$$

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