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It is given in the book that the Dirichlet's kernel $D_n(t) = 1/2 + \sum\limits_{k=1}^{n} \cos(kt)$ is given as $\frac{\sin(n+1/2)t}{2\sin(t/2)}$. I'd like to know if there is any such expression for $1/2 + \sum\limits_{k=1}^{n} \sin(kt)$ which is obtained by giving a $\pi/2$ shift for terms $D_n$.

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It is more natural to consider this sum without $1/2$. It will be a conjugate function to the Dirichlet's kernel: $$ \sum _{k=1}^n \sin (k x)=\csc \left(\frac{x}{2}\right) \sin \left(\frac{n x}{2}\right) \sin \left(\frac{1}{2} (n+1) x\right). $$

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