Answer
$$\int^{\pi/2}_{-\pi/2}\cos x\cos 7xdx=0$$
Work Step by Step
$$A=\int^{\pi/2}_{-\pi/2}\cos x\cos 7xdx$$
Apply the identity of the product of 2 cosine functions: $$\cos mx\cos nx=\frac{1}{2}[\cos(m-n)x+\cos(m+n)x]$$
we have $$A=\frac{1}{2}\int^{\pi/2}_{-\pi/2}\Big[\cos(-6x)+\cos8x\Big]dx$$
Since $\cos(-x)=\cos x$, $$A=\frac{1}{2}\int^{\pi/2}_{-\pi/2}(\cos6x+\cos8x)dx$$ $$A=\frac{1}{2}\Big(\frac{1}{6}\sin6x+\frac{1}{8}\sin8x\Big)\Big]^{\pi/2}_{-\pi/2}$$ $$A=\Big(\frac{1}{12}\sin6x+\frac{1}{16}\sin8x\Big)\Big]^{\pi/2}_{-\pi/2}$$ $$A=\frac{1}{12}\sin3\pi+\frac{1}{16}\sin4\pi-\Big(\frac{1}{12}\sin(-3\pi)+\frac{1}{16}\sin(-4\pi)\Big)$$ $$A=0+0-(0+0)=0$$
