Question

доказать тождество:

sin a + cos a - 1 = два корня из двух * sin a/2 * cos ( пи/4 + a/2) 

Answer 1

$sinalpha + cosalpha - 1 = 2sqrt2cdotsinfrac{alpha}2cdotcos left(frac{pi}4 +frac{alpha}2right)\cos left(frac{pi}4 +frac{alpha}2right)=cosfrac{pi}4cosfrac{alpha}2-sinfrac{pi}4sinfrac{alpha}2=frac{sqrt2}2cosfrac{alpha}2-frac{sqrt2}2sinfrac{alpha}2=\=frac{sqrt2}2(cosfrac{alpha}2-sinfrac{alpha}2)$

$2sqrt2cdotsinfrac{alpha}2cdotcos left(frac{pi}4 +frac{alpha}2right)=2sqrt2cdotfrac{sqrt2}2cdotsinfrac{alpha}2cdotleft(cosfrac{alpha}2-sinfrac{alpha}2right)=\=2sinfrac{alpha}2cosfrac{alpha}2-2sin^2frac{alpha}2=sinalpha-2sin^2frac{alpha}2\sinfrac{alpha}2=sqrt{frac{1-cosalpha}2}Rightarrow2sin^2frac{alpha}2=2cdotfrac{1-cosalpha}2=1-cosalpha\sinalpha-2sin^2frac{alpha}2=sinalpha+cosalpha-1$

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