Answer
$$\sin\frac{\pi}{5}\cos\frac{3\pi}{10}+\cos\frac{\pi}{5}\sin\frac{3\pi}{10}=1$$
Work Step by Step
$$X=\sin\frac{\pi}{5}\cos\frac{3\pi}{10}+\cos\frac{\pi}{5}\sin\frac{3\pi}{10}$$
Recall the sine sum identity that
$$\sin A\cos B+\cos A\sin B=\sin(A+B)$$
Looking back at $X$, we find $X$ is indeed the above identity with $A=\frac{\pi}{5}$ and $B=\frac{3\pi}{10}$.
Therefore, we can rewrite $X$ as
$$X=\sin\Big(\frac{\pi}{5}+\frac{3\pi}{10}\Big)$$
$$X=\sin\frac{5\pi}{10}=\sin\frac{\pi}{2}$$
$$X=1$$
In conclusion, $$\sin\frac{\pi}{5}\cos\frac{3\pi}{10}+\cos\frac{\pi}{5}\sin\frac{3\pi}{10}=1$$
