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Prove that $\sin(2x)-\sin(3x)$ has no real roots

Prove that $\sin(2x)-\sin(3x)$ has no real roots.

A:

$\sin(3x)-\sin(2x)$ is a linear combination of $\sin(3x)-\sin x$ and $\sin x-\sin(-x)$ so both it’s real part and imaginary part are zero.

A:

\begin{align}
\sin(2x)-\sin(3x) &= \sin(2x)-\cos(2x)\sin(x) + \cos(2x)\sin(x) – \cos(3x)\sin(x) + \cos(3x)\sin(x) \\
&= \sin(2x) \left(1-\cos(2x)\right) + \cos(3x) \left(\sin(x) – \sin(2x)\right)
\end{align}
Observe that $0 \leq 1-\cos(2x) \leq 2$ and $0 \leq \sin(x) – \sin(2x) \leq 2$ and therefore
$$0 \leq \sin(2x) \left(1-\cos(2x)\right) + \cos(3x) \left(\sin(x) – \sin(2x)\right) \leq 2\sin(2x) + 2\cos(3x) \leq 4\cos^2(x/2) = 2$$
We conclude that the function is nonnegative.

Meditation

Meditation is widely used in Buddhist meditation practice as a form of Mindfulness. Like mindfulness, meditation is a skillful and deliberate technique for cultivating concentration (samadhi) on the present moment

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