## Trigonometry (11th Edition) Clone

$$\sin\Big(\frac{\pi}{4}+x\Big)=\frac{\sqrt2}{2}(\cos x+\sin x)$$
$$X=\sin\Big(\frac{\pi}{4}+x\Big)$$ 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, $$\sin\Big(\frac{\pi}{4}+x\Big)=\frac{\sqrt2}{2}(\cos x+\sin x)$$