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July 2022 1 10 Report

Докажите тождество:
sin2t/1+cos2t*cost/1+cost=tgt/2
sin2t/1+cos2t*cost/1+cost*cos1/2/1+cost t/2=tg1/4

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Verified answer

$1)~\boldsymbol{\dfrac{\sin(2t)}{1+\cos(2t)}\cdot\dfrac{\cos t}{1+\cos t}=}\dfrac{2\sin t\cos t}{2\cos^2t}\cdot\dfrac{\cos t}{1+\cos t}=\\\\\\~~~=\dfrac{\sin t}{1+\cos t}=\dfrac{2\sin \frac t2\cdot\cos \frac t2}{2\cos^2 \frac t2}=\dfrac{\sin \frac t2}{\cos \frac t2}=\bold{ \text{tg }\frac t2}~~~\blacksquare$

2) В условии в третьей и четвёртой дроби исправлены опечатки

$\boldsymbol{\dfrac{\sin(2t)}{1+\cos(2t)}\cdot\dfrac{\cos t}{1+\cos t}\cdot\dfrac{\cos \frac t2}{1+\cos \frac t2}=}\\\\\\=\dfrac{2\sin t\cos t}{2\cos^2t}\cdot\dfrac{\cos t}{1+\cos t}\cdot\dfrac{\cos \frac t2}{1+\cos \frac t2}=$

$=\dfrac{\sin t}{1+\cos t}\cdot\dfrac{\cos \frac t2}{1+\cos \frac t2}=\dfrac{2\sin \frac t2\cos \frac t2}{2\cos^2 \frac t2}\cdot\dfrac{\cos \frac t2}{1+\cos \frac t2}=\\\\\\=\dfrac{\sin \frac t2}{1+\cos \frac t2}=\dfrac{2\sin \frac t4\cos \frac t4}{2\cos^2 \frac t4}=\dfrac{\sin \frac t4}{\cos \frac t4}=\bold{ \text{tg }\frac t4}~~~\blacksquare$

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Использованы формулы двойного аргумента

$\sin (2\alpha )=2\sin \alpha \cos \alpha \\\cos(2\alpha )=2\cos^2\alpha -1\\1+\cos(2\alpha )=2\cos^2\alpha$

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