Question

# Show that: ( cos 2 theta cos frac{theta}{2}-cos 3 theta cos frac{9 theta}{2}=sin 5 theta sin frac{5 theta}{2} )

Solution

( 2 cos 2 theta cos frac{theta}{2}-cos 3 theta cos frac{9 theta}{2}=frac{-cos frac{3 theta}{2}+cos frac{50}{2}-cos frac{150}{2}+cos 3 theta}{2} )

( =frac{cos frac{50}{2}-cos frac{150}{2}}{2}=frac{2}{2} sin left(frac{150-frac{50}{2}}{2}right) sin left(frac{150+frac{50}{2}}{2}right) )

( =sin left(frac{10 theta}{4}right) sin left(frac{20 theta}{4}right)=sin (5 theta) sin left(frac{5 theta}{2}right) )