Question

cho x+y+z+t=2\(\Pi\)

CMR \(\cos^2x+\cos^2y-\cos^2z-\cos^2t=-2\sin\left(x+y\right)\sin\left(y+z\right)\cos\left(x+z\right)\)

Answer 1

\(cos^2t=cos^2\left(2\pi-x-y-z\right)=cos^2\left(x+y+z\right)\)

\(VT=cos^2x+cos^2y-cos^2z-cos^2\left(x+y+z\right)\)

\(=\frac{1}{2}+\frac{1}{2}cos2x+\frac{1}{2}+\frac{1}{2}cos2y-\frac{1}{2}-\frac{1}{2}cos2z-\frac{1}{2}-\frac{1}{2}cos\left(2x+2y+2z\right)\)

\(=\frac{1}{2}\left(cos2x-cos2z\right)+\frac{1}{2}\left(cos2y-cos\left(2x+2y+2z\right)\right)\)

\(=-sin\left(x+z\right)sin\left(x-z\right)-sin\left(x+2y+z\right)sin\left(-x-z\right)\)

\(=sin\left(x+z\right)\left[sin\left(x+2y+z\right)-sin\left(x-z\right)\right]\)

\(=2sin\left(x+z\right).sin\left(y+z\right).cos\left(x+y\right)\)

Không giống :D