17. $\sin ^{3} x+\cos ^{3} x+\sin \left(x+\frac{\pi}{4}\right)=0$.

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17. $\sin ^{3} x+\cos ^{3} x+\sin \left(x+\frac{\pi}{4}\right)=0$.
17. Factoring the sum of cubes and replacing the sum $\sin x+\cos x$ with a product, we can transform the equation into the following form:

$$
\cos \left(x-\frac{\pi}{4}\right)(\sqrt{2}-\sqrt{2} \sin x \cos x+1)=0
$$

From which $\cos \left(x-\frac{\pi}{4}\right)=0, \quad x=\frac{3}{4} \pi+\pi k$.

The second factor does not become zero, since $\sin 2 x \neq 2+\sqrt{2}$.