$$\sin^{2}(\theta)+\cos^{2}(\theta)=1$$ $$\Rightarrow \sin^{2}(\alpha)+\cos^{2}(\alpha)=1$$ $$\Rightarrow (\frac{15}{17})^{2}+\cos^{2}(\alpha)=1$$ $$\Rightarrow \frac{225}{289}+\cos^{2}(\alpha)=1$$ $$\Rightarrow \cos^{2}(\alpha)=\frac{64}{289}$$ $$\Rightarrow \cos(\alpha)=\sqrt{\frac{64}{289}}=\frac{8}{17}$$