Let P = : -=√2 and Q = : +=√2 be two sets.

Question

Let $P = \{	heta : \sin	heta-\cos	heta=\sqrt2\cos	heta\}$ and $Q = \{	heta : \sin	heta+\cos	heta=\sqrt2\sin	heta\}$ be two sets. Then:

Accepted Answer

From $P$'s equation: $\sin\theta = (1+\sqrt2)\cos\theta$.

Since $1+\sqrt2 = \dfrac{1}{\sqrt2-1}$, this gives $(\sqrt2-1)\sin\theta=\cos\theta$, which rearranges to:
$$\sqrt2\sin\theta = \sin\theta+\cos\theta$$

This is exactly $Q$'s defining equation. So the two conditions are algebraically equivalent: $P=Q$.

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