The sum of angles formula for sine is $\sin(a+b) = \sin a \cos b + \cos a \sin b$ and for cosine is $\cos(a+b) = \cos a \cos b - \sin a \sin b$. Applying these formulas, we get:
$$\sin \left(t+\frac{\pi}{4}\right)+\cos \left(t+\frac{\pi}{4}\right) = \sin t \cos
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