Answer
$$\sin\Big(\frac{\pi}{4}+x\Big)=\frac{\sqrt2}{2}(\cos x+\sin x)$$
Work Step by Step
$$X=\sin\Big(\frac{\pi}{4}+x\Big)$$
According to sine sum identity:
$$\sin(A+B)=\sin A\cos B+\sin B\cos A$$
That means
$$X=\sin\frac{\pi}{4}\cos x+\sin x\cos\frac{\pi}{4}$$
$$X=\frac{\sqrt2}{2}\cos x+\frac{\sqrt2}{2}\sin x$$
$$X=\frac{\sqrt2}{2}(\cos x+\sin x)$$
Overall,
$$\sin\Big(\frac{\pi}{4}+x\Big)=\frac{\sqrt2}{2}(\cos x+\sin x)$$
