Question

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

Answer 1

`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=(\pi)/(2)+k\pi\ \ \ \ (k\in\mathbb{Z})`


`For\ \theta=k\pi\to cosk\pi-sink\pi=\pm1-0=\pm1\to conclusion:\theta=2k\pi\\\\For\ \theta=(\pi)/(2)+k\pi\to cos((\pi)/(2)+k\pi)-sin((\pi)/(2)+k\pi)=0-(\pm1)=\mp1\to\\\to conclusion:\theta=(3\pi)/(2)+2k\pi\\\\============================\\\\cos\theta+sin\theta=(*)\\\\for\ \theta=2k\pi\to(*)=1+0=1\\\\for\ \theta=(3\pi)/(2)+2k\pi\to(*)=0-1=-1`