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if cos θ -sin θ =1, find θ cos θ +sin θ?

Sagot :

[tex]cos\theta-sin\theta=1\ \ \ \ |square\ both\ sides\\\\(cos\theta-sin\theta)^2=1\\\\cos^2\theta-2sin\theta cos\theta+sin^2\theta=1\\\\1-2sin\theta cos\theta=1\\\\-2sin\theta cos\theta=1-1\\\\-2sin\theta cos\theta=0\iff sin\theta=0\ or\ cos\theta=0\\\\\theta=k\pi\ or\ \theta=\frac{\pi}{2}+k\pi\ \ \ \ (k\in\mathbb{Z})[/tex]

[tex]For\ \theta=k\pi\to cosk\pi-sink\pi=\pm1-0=\pm1\to conclusion:\theta=2k\pi\\\\For\ \theta=\frac{\pi}{2}+k\pi\to cos(\frac{\pi}{2}+k\pi)-sin(\frac{\pi}{2}+k\pi)=0-(\pm1)=\mp1\to\\\to conclusion:\theta=\frac{3\pi}{2}+2k\pi\\\\============================\\\\cos\theta+sin\theta=(*)\\\\for\ \theta=2k\pi\to(*)=1+0=1\\\\for\ \theta=\frac{3\pi}{2}+2k\pi\to(*)=0-1=-1[/tex]
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