$f\left(x\right)=\frac{\sin x+\sin 3x+\sin 5x+\dots +\sin 15x}{\cos x+\cos 3x+\cos 5x+\dots +\cos 15x}\text{ then the value of }f\left(x\right)\text{ at }x=\frac{\pi }{32}$

$\begin{array}{l}\sin \alpha +\sin (\alpha +d)+\dots .+\sin \left(\alpha +(n-1)d\right)==\frac{\sin \left(\frac{nd}{2}\right)}{\sin \frac{d}{2}}\sin \left(\frac{\alpha +\alpha +(n-1)d}{2}\right)\\ and\\ \cos \alpha +\cos (\alpha +d)+\dots .+\cos (\alpha +(n-1\left)d\right)==\frac{\sin \frac{nd}{2}}{\sin \frac{d}{2}}\cos \frac{(\alpha +\alpha +(n-1\left)\right)d)}{2}\\ \therefore \frac{\sin x+\sin 3x+\dots .+\sin 15x}{\cos x+\cos 3x+\dots .+\cos 15x}=\frac{\sin \left(\frac{x+15x}{2}\right)}{\cos \left(\frac{x+15x}{2}\right)}=\tan 8x\\ \end{array}$

$\therefore f\left(\frac{\pi }{32}\right)=\tan 8\left(\frac{\pi }{32}\right)=1$