Question

Chứng minh đẳng thức:

\(\dfrac{2sin2x+sin4x}{2\left(cosx+cos3x\right)}\)=\(tan2x.cosx\)

Answer 1

\(VT=\dfrac{2\cdot sin2x+2\cdot sin2x\cdot cos2x}{2\cdot\left(cosx+cos3x\right)}\)

\(=\dfrac{2\cdot sin2x\left(1+cos2x\right)}{2\cdot\left(cosx+cos3x\right)}\)

\(=\dfrac{sin2x\cdot\left(1+2cos^2x-1\right)}{cosx+cos3x}\)

\(=\dfrac{sin2x\cdot2\cdot cos^2x}{2\cdot cos\left(\dfrac{3x+x}{2}\right)\cdot cos\left(\dfrac{3x-x}{2}\right)}\)

\(=\dfrac{sin2x\cdot cos^2x}{cosx\cdot cos2x}=\dfrac{sin2x}{cos2x}\cdot cosx=tan2x\cdot cosx\)