Question

выполнить номер пятьсот

Answer 1

1)Формула
$tg alpha +tg beta = frac{sin ( alpha + beta )}{cos alpha cos beta }$
$tg(x+ frac{ pi }{4}) +tg (x- frac{ pi }{4} )= frac{sin (x+ frac{ pi }{4}+x- frac{ pi }{4} )}{cos (x+ frac{ pi }{4}) cos (x- frac{ pi }{4}) }= frac{sin 2x }{cos (x+ frac{ pi }{4}) cos (x- frac{ pi }{4}) }$

2)$cos2 alpha +sin2 alpha tg alpha=cos2 alpha+ 2sin alpha cos alphafrac{sin alpha }{cos alpha } =cos2 alpha +2sin ^{2} alpha = \ =cos ^{2} alpha - sin ^{2} alpha +2sin ^{2} alpha =cos ^{2} alpha +sin ^{2} alpha =1$

3)2+tgα+ctgα=
$=2+ frac{sin 2alpha }{cos2 alpha }+ frac{cos2 alpha }{sin2 alpha }= frac{2cos2 alpha cdot sin2 alpha +sin ^{2}2 alpha +cos ^{2}2 alpha }{cos2 alpha cdot sin2 alpha } = frac{(sin2 alpha +cos2 alpha ) ^{2} }{sin2 alpha cos2 alpha }$

4)1-0,25 sin²2α-cos²2α-cos⁴α=0,25(4- sin²2α-4cos²2α-4cos⁴α)=
=0,25(4cos²2α+4sin²2α- sin²2α-4cos²2α-4cos⁴α)=0,25(3sin²2α-4cos⁴α)=
=0,25(3·4sin⁴αcos⁴α-4cos⁴α)=0,25·4cos⁴α·(sin⁴α-1)=cos⁴α·(sin⁴α-1)