Question

(1-2cos(x/2)+cosx)/(1+2cos(x/2)+cosx) = ?

Answer 1

$frac{1-2cos frac{x}{2}+cosx }{1+2cos frac{x}{2}+cosx } = frac{1+cosx-2cos frac{x}{2} }{1+cosx+2cos frac{x}{2} } = frac{2cos^{2} frac{x}{2}-2cos frac{x}{2} }{2cos^{2} frac{x}{2}+2cos frac{x}{2} } = \ frac{2cos frac{x}{2}(cos frac{x}{2} -1) }{2cos frac{x}{2}(cos frac{x}{2}+1) }= \ frac{-2sin^{2} frac{x}{4} }{2cos^{2} frac{x}{4} } =-tg^{2} frac{x}{4}$