Question

помогите доказать тождество​

Answer 1

$\frac{\cos^2(a) - \sin^2(a)}{1 + 2\sin(a)\cos(a)} =$

$= \frac{(\cos(a) - \sin(a))\cdot(\cos(a) + \sin(a))}{\cos^2(a) + \sin^2(a) + 2\sin(a)\cos(a)} =$

$= \frac{(\cos(a) - \sin(a))\cdot(\cos(a) + \sin(a))}{(\cos(a) + \sin(a))^2} =$

$= \frac{\cos(a) - \sin(a)}{\cos(a) + \sin(a)} =$

$= \frac{\frac{\cos(a) - \sin(a)}{\cos(a)}}{\frac{\cos(a) + \sin(a)}{\cos(a)}} =$

$= \frac{1 - \frac{\sin(a)}{\cos(a)}}{1 + \frac{\sin(a)}{\cos(a)}} =$

$= \frac{1 - \mathrm{tg}(a)}{1 + \mathrm{tg}(a)}$

Answer 2

Объяснение:

$\frac{cos^2\alpha -sin^2\alpha }{1+2*sin\alpha *cos\alpha \alpha}=\frac{cos^2\alpha -sin^2\alpha }{sin^2\alpha +cos^2\alpha +2*sin\alpha *cos\alpha } =\frac{cos^2\alpha -sin^2\alpha }{sin^2\alpha +2*sin\alpha *cos\alpha +cos^2\alpha }=\frac{cos^2\alpha -sin^2\alpha }{(sin\alpha +cos\alpha )^2} .$

Разделим одновременно числитель и знаменатель на cos²α:

$\frac{1-tg^2\alpha }{\frac{(sin\alpha +cos\alpha )^2}{cos^2\alpha } } =\frac{1-tg^2\alpha }{(\frac{sin\alpha +cos\alpha }{cos\alpha } )^2} =\frac{(1+tg\alpha )*(1-tg\alpha )}{(1+tg\alpha )^2} =\frac{1-tg\alpha }{1+tg\alpha } .$