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I have tried to solve this question (4.2.3). I got the magnitude spectrum right but the phase spectrum wrong. I am getting the phase spectrum at $ \frac{\pi}{4} $ and $ - \frac{\pi}{4} $. However, in the solution manual, they have plotted it at $ \frac{\pi}{2} $ and $ - \frac{\pi}{2} $.

The phase is visibly at $\frac{\pi}{4} $ and $ - \frac{\pi}{4} $ from the coefficients $ e^{- \frac{j\pi}{4} }$ and $ e^{ \frac{j\pi}{4} } $ in the two terms.

I am not able to understand why am I wrong and why are they right!

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