Câu 5:
a: \(M=cos^2a+cos^2\left(\frac{2\pi}{3}-a\right)+cos^2\left(\frac{2\pi}{3}+a\right)\)
\(=\frac{1+cos2a}{2}+\frac{1+cos\left\lbrack2\cdot\left(\frac23\pi-a\right)\right\rbrack}{2}+\frac{1+cos\left\lbrack2\cdot\left(\frac23\pi+a\right)\right\rbrack}{2}\)
\(=\frac{1+cos2a}{2}+\frac{1+cos\left(\frac43\pi-2a\right)}{2}+\frac{1+cos\left(\frac43\pi+2a\right)}{2}\)
\(=\frac32+\frac12\cdot cos2a+\frac12\cdot\left\lbrack cos\left(\frac43\pi-2a\right)+cos\left(\frac43\pi+2a\right)\right\rbrack\)
\(=\frac32+\frac12\cdot cos2a+\frac12\cdot2\cdot cos\left(\frac{\frac43\pi-2a+\frac43\pi+2a}{2}\right)\cdot cos\left(\frac{\frac43\pi-2a-\frac43\pi-2a}{2}\right)\)
\(=\frac32+\frac12\cdot cos2a+cos\left(\frac43\pi\right)\cdot cos\left(-2a\right)=\frac32+\frac12\cdot cos2a+\left(-\frac12\right)\cdot cos2a=\frac32\)
b: \(N=cos\left(a-\frac{\pi}{3}\right)\cdot cos\left(a+\frac{\pi}{4}\right)+cos\left(a+\frac{\pi}{6}\right)\cdot cos\left(a+\frac34\pi\right)\)
\(=\frac12\cdot cos\left(a-\frac{\pi}{3}-a-\frac{\pi}{4}\right)\cdot cos\left(a-\frac{\pi}{3}+a+\frac{\pi}{4}\right)+\frac12\cdot cos\left(a+\frac{\pi}{6}-a-\frac34\pi\right)\cdot cos\left(a+\frac{\pi}{6}+a+\frac34\pi\right)\)
\(=\frac12\cdot cos\left(-\frac{7}{12}\pi\right)\cdot cos\left(2a-\frac{\pi}{12}\right)+\frac12\cdot cos\left(-\frac{7}{12}\pi\right)\cdot cos\left(2a+\frac{11}{12}\pi\right)\)
\(=\frac12\cdot cos\left(\frac{7}{12}\pi\right)\cdot cos\left(2a-\frac{\pi}{12}\right)+\frac12\cdot cos\left(\frac{7}{12}\pi\right)\cdot cos\left(2a-\frac{\pi}{12}+\pi\right)\)
\(=\frac12\cdot cos\left(\frac{7}{12}\pi\right)\cdot cos\left(2a-\frac{\pi}{12}\right)+\frac12\cdot cos\left(\frac{7}{12}\pi\right)\cdot\left(-1\right)\cdot cos\left(2a-\frac{\pi}{12}\right)\)
=0
