Question

Доказать тождество:
1) sin2x-tgx=cos2x*tgx
2)tgx+2ctg2x=ctgx

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Answer 1

Доказать тождество:

$sin (2x)-tg (x) = cos (2x)\cdot tg (x))\\sin (2x)-tg (x) = \left(2cos ^2(x)-1\right)tg (x)\\sin (2x)-tg (x) = 2cos^2(x)tg(x)-tg (x)\\sin (2x)-tg (x) = 2cos ^2(x)\frac{sin (x)}{cos (x)}-tg (x)\\sin (2x)-tg (x) = 2cos (x)sin (x)-tg (x)\\sin (2x)-tg (x) = \sin \left(2x\right)-tg (x)$

тождество доказано

$tg (x)+2ctg (2x)=ctg (x)\\\\\frac{sin(x)}{cos(x)}+2\cdot \frac{cos(2x)}{sin(2x)}=ctg(x)\\\\\frac{sin(2x)sin(x)+2cos(2x)cos(x)}{cos(x)sin(2x)}=ctg(x)\\\\\frac{2cos(2x)cos(x)+2cos(x)sin(x)sin(x)}{2cos(x)cos(x)sin(x)}=ctg(x)\\\\\frac{cos(2x)+sin^2(x)}{cos(x)sin(x)}=ctg(x)\\\\\frac{cos^2(x)-sin^2(x)+sin^2(x)}{cos(x)sin(x)}=ctg(x)\\\\\frac{cos^2(x)}{cos(x)\sin(x)}=ctg(x)\\\\\frac{cos(x)}{sin(x)}=ctg(x)\\\\ctg(x) = ctg(x)$

тождество доказано