Question

Докажите тождество.
Интересно кто за это возьмётся)

Answer 1

$1)\sqrt{\frac{\frac{tg^{2}x }{Ctg^{2} x}*Cos^{5}x}{Ctgx*Sinx} }=\sqrt{\frac{\frac{Sin^{2}x*Sin^{2}x}{Cos^{2}x*Cos^{2}x}*Cos^{5}x }{\frac{Cosx}{Sinx}*Sinx } }=\sqrt{\frac{Sin^{4}x*Cosx }{Cosx} }=\sqrt{Sin^{4} }x=Sin^{2}x\\\\\\2)\sqrt{\frac{(\frac{tg^{2}}{Ctg^{2} x})^{-1}*Sin^{5} x }{tgx*Cosx } }=\sqrt{\frac{\frac{Cos^{2}x*Cos^{2} x }{Sin^{2}x*Sin^{2}x}*Sin^{5}x}{\frac{Sinx}{Cosx} *Cosx} }=\sqrt{\frac{Cos^{4}x*Sinx }{Sinx} }=\sqrt{Cos^{4}x }=Cos^{2}x$

$3)(\sqrt{tgx(\sqrt{\frac{Ctgx}{Cosx} )^{-2} } })^{-2} =(\sqrt{tgx*\frac{Cosx}{Ctgx} } )^{-2}=(\sqrt{\frac{Sinx}{Cosx}*\frac{Cosx*Sinx}{Cosx}})^{-2}=\frac{Cosx }{Sin^{2}x } \\\\\\4)(\frac{Cosx}{Sin^{2}x }*tg^{2}x*Cos^{2}x)^{2}=(\frac{Cosx}{Sin^{2}x }*\frac{Sin^{2}x }{Cos^{2}x }*Cos^{2}x)^{2}=Cos^{2}x\\\\\\5)\frac{Sin^{2}x+Cos^{2}x}{Cos^{2}x }=\frac{Sin^{2}x }{Cos^{2} x} +\frac{Cos^{2}x }{Cos^{2}x } =tg^{2}x+1$

$tg^{2} x+1=tg^{2} x+1$

Тождество доказано