Question

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

Answer 1

$\sin\alpha + \cos\alpha - 1 = 2\sqrt2\cdot\sin\frac{\alpha}2\cdot\cos \left(\frac{\pi}4 +\frac{\alpha}2\right)\\\cos \left(\frac{\pi}4 +\frac{\alpha}2\right)=\cos\frac{\pi}4\cos\frac{\alpha}2-\sin\frac{\pi}4\sin\frac{\alpha}2=\frac{\sqrt2}2\cos\frac{\alpha}2-\frac{\sqrt2}2\sin\frac{\alpha}2=\\=\frac{\sqrt2}2(\cos\frac{\alpha}2-\sin\frac{\alpha}2)$

$2\sqrt2\cdot\sin\frac{\alpha}2\cdot\cos \left(\frac{\pi}4 +\frac{\alpha}2\right)=2\sqrt2\cdot\frac{\sqrt2}2\cdot\sin\frac{\alpha}2\cdot\left(\cos\frac{\alpha}2-\sin\frac{\alpha}2\right)=\\=2\sin\frac{\alpha}2\cos\frac{\alpha}2-2\sin^2\frac{\alpha}2=\sin\alpha-2\sin^2\frac{\alpha}2\\\sin\frac{\alpha}2=\sqrt{\frac{1-\cos\alpha}2}\Rightarrow\2\sin^2\frac{\alpha}2=2\cdot\frac{1-\cos\alpha}2=1-\cos\alpha\\\sin\alpha-2\sin^2\frac{\alpha}2=\sin\alpha+\cos\alpha-1$

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