13.

\(\dfrac{sinx-sin3x+sin5x}{cosx-cos3x+cos5x}\)

\(=\dfrac{2sin3x.cos2x-sin3x}{2cos3x.cos2x-cos3x}\)

\(=\dfrac{\left(2cos2x-1\right)sin3x}{\left(2cos2x-1\right)cos3x}\)

\(=\dfrac{sin3x}{cos3x}=tan3x\)

14)\(\dfrac{1-sin2x}{1+sin2x}=\dfrac{sin\dfrac{\pi}{4}-sin2x}{sin\dfrac{\pi}{4}+sin2x}\)\(=\dfrac{2.sin\left(\dfrac{\pi}{4}-x\right)cos\left(\dfrac{\pi}{4}+x\right)}{2sin\left(\dfrac{\pi}{4}+x\right)cos\left(\dfrac{\pi}{4}-x\right)}\)\(=tan\left(\dfrac{\pi}{4}-x\right)cot\left(\dfrac{\pi}{4}+x\right)\)\(=tan\left[\dfrac{\pi}{2}-\left(\dfrac{\pi}{4}+x\right)\right]cot\left(\dfrac{\pi}{4}+x\right)\)

\(=cot\left(\dfrac{\pi}{4}+x\right).cot\left(\dfrac{\pi}{4}+x\right)\)

\(=cot^2\left(\dfrac{\pi}{4}+x\right)\)

15)\(\dfrac{sinx+sin3x+sin5x}{cosx+cos3x+cos5x}\)\(=\dfrac{\left(sinx+sin5x\right)+sin3x}{\left(cosx+cos5x\right)+cos3x}\)\(=\dfrac{2sin3x.cos\left(-2x\right)+sin3x}{2cos3x.cos\left(-2x\right)+cos3x}=\dfrac{sin3x\left[2cos\left(-2x\right)+1\right]}{cos3x\left[2cos\left(-2x\right)+1\right]}\)\(=tan3x\)

16)\(\dfrac{cos5x-cosx}{sin4x+sin2x}=\dfrac{-2.sin3x.sin2x}{2.sin3x.cosx}\)\(=\dfrac{-2.sinx.cosx}{cosx}\)\(=-2.sinx\)

17)\(\dfrac{sin^4x-cos^4x+cos^2x}{2\left(1-cosx\right)}\)\(=\dfrac{cos^2x-\left(sin^2x+cos^2x\right)\left(cos^2x-sin^2x\right)}{2\left(1-cosx\right)}\)\(=\dfrac{cos^2x-\left(cos^2x-sin^2x\right)}{2\left(1-cosx\right)}\)\(=\dfrac{sin^2x}{2\left(1-cosx\right)}\)

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

18) (Xem lại đề)

19)\(\dfrac{1+cosx+cos2x+cos3x}{2cos^2x+cosx-1}\)\(=\dfrac{\left(1+cos2x\right)+\left(cosx+cos3x\right)}{2cos^2x+cosx-1}\)\(=\dfrac{2cos^2x+2.cos2x.cosx}{2cos^2x+cosx-1}\)

\(=\dfrac{2cosx\left(cosx+cos2x\right)}{2cos^2x+cosx-1}\)\(=\dfrac{2cosx\left(cosx+2cos^2-1\right)}{2cos^2x+cosx-1}\)\(=2cosx\)