Question

Chứng minh đẳng thức sau: \(\dfrac{3}{4}-\cos^2\left(a-\dfrac{\pi}{3}\right)+\cos a.\cos\left(a-\dfrac{\pi}{3}\right)=\cos^2a\)

Answer 1

\(VT=\dfrac{3}{4}-\dfrac{1}{2}-\dfrac{1}{2}cos\left(2a-\dfrac{2\pi}{3}\right)+\dfrac{1}{2}cos\left(2a-\dfrac{\pi}{3}\right)+\dfrac{1}{2}cos\left(\dfrac{\pi}{3}\right)\)

\(=\dfrac{1}{2}+\dfrac{1}{2}\left[cos\left(2a-\dfrac{\pi}{3}\right)-cos\left(2a-\dfrac{2\pi}{3}\right)\right]\)

\(=\dfrac{1}{2}-sin\left(2a-\dfrac{\pi}{2}\right)sin\left(\dfrac{\pi}{6}\right)\)

\(=\dfrac{1}{2}+\dfrac{1}{2}cos2a=\dfrac{1}{2}+\dfrac{1}{2}\left(2cos^2a-1\right)=cos^2a\)