Question

tg(5n/4+x)+tg(5n/4-x)​

Answer 1

$tg(frac{5pi }{4}+x)+tg(frac{5pi }{4}-x)=frac{sinfrac{5pi }{2}}{cos(frac{5pi }{4}+x)cdot cos(frac{5pi }{4}-x)}=\\=frac{1}{frac{1}{2}cdot (cosfrac{5pi }{2}+cos2x)}}=frac{2}{1+cos2x}=frac{2}{2cos^2x}=frac{1}{cos^2x}=1+tg^2x\\\\P.S.; ; ; ; tg(alpha +beta )=frac{sin(alpha +beta )}{cosalpha cdot cosbeta }; ; ; ,; ; ; cos^2alpha =frac{1+cos2alpha }{2}; ; ,\\1+tg^2alpha =frac{1}{cos^2alpha }$