Question

Докажите тождество плз.

Answer 1

$frac{Sin alpha }{1-Cos alpha }- frac{1+Cos alpha }{Sin alpha }= frac{Sin ^{2} alpha -1+Cos ^{2} alpha }{Sin alpha (1-Cos alpha) }= frac{1-1}{Sin alpha (1-Cos alpha )}= frac{0}{Sin alpha (1-Cos alpha )}=0$
Разность правой и левой частей равна нулю, значит тождество доказано.

$frac{1-(Sin alpha +Cos alpha ) ^{2} }{Sin alpha Cos alpha -Ctg alpha }= frac{1-(Sin ^{2} alpha+2Sin alpha Cos alpha +Cos ^{2} alpha ) }{Sin alpha Cos alpha - frac{Cos alpha }{Sin alpha } }= frac{1-1-2Sin alpha Cos alpha }{ frac{Sin ^{2} alpha Cos alpha -Cos alpha }{Sin alpha } }=$$frac{-2Sin ^{2} alpha Cos alpha }{Cos alpha (Sin ^{2} alpha -1)} = frac{-2Sin ^{2} alpha }{-Cos ^{2} alpha } =2tg ^{2} alpha$

Answer 2

a
sina/(1-cosa)=2sin(a/2)cos(a/2)/2sin²(a/2)=ctg(a/2)
(1+cosa)/sina=(2cos²(a/2)/2sin(a/2)cos(a/2)=ctg(a/2)
ctg(a/2)=ctg(a/2)
b
(sin²a+cos²a-sin²a-2sinacosa-cos²a)/(sinacosa-cosa/sina)=
=-2sinacosa*sina/(sin²acosa-cosa)=(-2sin²acosa)/cosa(sin²a-1)=
=(-2sin²acosa)/(-cos³a)=2tg²a