Question

Решите примеры по тригонометрии, : )

Answer 1

$g)\frac{4Sin^{4}\frac{\alpha }{4}}{1-Cos^{2} \frac{\alpha }{2} }=\frac{4Sin^{4}\frac{\alpha }{4}}{Sin^{2}\frac{\alpha }{2}}=\frac{4Sin^{4}\frac{\alpha }{4}}{(2Sin\frac{\alpha }{4} Cos\frac{\alpha }{4} )^{2} }=\frac{4Sin^{4}\frac{\alpha }{4}}{4Sin^{2}\frac{\alpha }{4}Cos^{2}\frac{\alpha }{4}}=\frac{Sin^{2}\frac{\alpha }{4}}{Cos^{2}\frac{\alpha }{4}}=tg^{2}\frac{\alpha }{4}\\\\tg^{2}\frac{\alpha }{4}=tg^{2}\frac{\alpha }{4}$

Тождество доказано

$i)Cos3xCos6xCos12x=\frac{Sin24x}{8Sin3x}\\\\8Sin3x*Cos3xCos6xCos12x=8Sin3x*\frac{Sin24x}{8Sin3x}\\\\(2Sin3xCos3x)*4Cos6xCos12x=Sin24x\\\\Sin6x*4Cos6xCos12x=Sin24x\\\\(2Sin6xCos6x)*2Cos12x=Sin24x\\\\Sin12x*2Cos12x=Sin24x\\\\Sin24x=Sin24x$

Тождество доказано

$k)tg(\alpha+\frac{\pi }{4})+tg(\alpha \frac{\pi }{4})=\frac{Sin(\alpha+\frac{\pi }{4}+\alpha-\frac{\pi }{4})}{Cos(\alpha+\frac{\pi }{4})*Cos(\alpha-\frac{\pi }{4})}=\frac{Sin2\alpha }{\frac{1}{2}(Cos(\alpha+\frac{\pi }{4}+\alpha-\frac{\pi }{4})+Cos(\alpha+\frac{\pi }{4}-\alpha+\frac{\pi }{4})}=\frac{2Sin2\alpha }{Cos2\alpha+Cos\frac{\pi }{2}} =\frac{2Sin2\alpha }{Cos2\alpha+0 }=\frac{2Sin2\alpha }{Cos2\alpha }=2tg2\alpha\\\\2tg2\alpha=2tg2\alpha$

Тождество доказано