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Please help me out in answering this:

Show that $|\sin x−\cos x|≤ 2$ for all $x.$

I don't know where to start and I think we have to use the mean value theorem to show this.

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closed as off-topic by José Carlos Santos, Arnaud D., Guy Fsone, Brian Borchers, daw Nov 9 '17 at 17:57

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    $\begingroup$ Here's a better problem: prove that $|\sin x-\cos x|\le\sqrt2$ for all $x$. $\endgroup$ – Lord Shark the Unknown Nov 9 '17 at 6:55
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    $\begingroup$ Have you heard of the triangle inequality? $\endgroup$ – luthien Nov 9 '17 at 6:56
  • $\begingroup$ Nope haven't heard of it. $\endgroup$ – hel Nov 9 '17 at 7:02
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    $\begingroup$ The triangle inequality states that $|a+b| \leq |a|+|b|$. In this case you can treat $\sin(x)$ as $a$ and $-\cos(x)$ as $b$. Then consider the range of $\cos(x)$ and $\sin(x)$. What are the maximum values that they can take? $\endgroup$ – luthien Nov 9 '17 at 7:04
  • $\begingroup$ @hel Start with right triangle having sides $1,\cos x, \sin x $ $\endgroup$ – Narasimham Nov 9 '17 at 17:31
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We will use the fact for all $x \in \mathbb{R}$, $|\sin x|, |\cos x| \le 1$.

Method 1 - Triangle inequality ( $|a + b| \le |a| + |b|$ for all $a, b \in \mathbb{R}$)

$$|\sin(x) - \cos(x)| \le |\sin(x)| + |\cos(x)| \le 1 + 1 = 2$$ Method 2 - difference formula for sine ( $\sin(\theta-\phi) = \sin\theta\cos\phi - \cos\theta\sin\phi$ for all $\theta, \phi \in \mathbb{R}$). $$|\sin(x)-\cos(x)| = \sqrt{2}\left|\sin(x)\cos\frac{\pi}{4} -\cos(x)\sin\frac{\pi}{4}\right| = \sqrt{2}\left|\sin\left(x - \frac{\pi}{4}\right)\right| \le \sqrt{2} < 2$$

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Notice that $\sin (x)$ and $\cos (x)$ are bounded by $1$, and use Triangle Inequality. Therefore you will get the wanted result

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If you do not know where to start then ...start from determining the range of values $\sin x$ and $\cos x$ can take.

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Guide:

$$-1 \leq \sin(x) \leq 1$$ $$-1 \leq \cos(x) \leq 1$$

  • Work out the range that $-\cos(x)$ can take.
  • Work out the range that $\sin(x) - \cos(x)$ can take.
  • Work out the range that $|\sin(x) - \cos(x)|$ can take.
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Note: $$|\sin x-\cos x|\le 2 \iff -2\le \sin x-\cos x\le 2 \iff$$ $$-2+\cos x\le \sin x\le 2+\cos x \iff $$ $$-2+\cos x\le -1\le \sin x\le 1\le 2+\cos x.$$

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I have another method for you if you want to use some straight-up calculus optimization. Let us first determine the extrema of $\sin(x)-\cos(x)$. To do this, we take the derivative and set it equal to $0$. In this case, the derivative is $\cos(x) +\sin(x)$, so we want to solve $\cos(x) + \sin(x) = 0$, or $\cos(x) = -\sin(x)$.

So when are $\sin(x)$ and $\cos(x)$ negatives of each other? This occurs precisely when $x = \frac{3\pi}{4}$ or $x= -\frac{\pi}{4}$. This means our extrema are at these values of $x$.

Let's now classify these extrema. First plug in $x = \frac{3\pi}{4}$ to $\sin(x)-\cos(x)$: $$\sin\bigg(\frac{3\pi}{4}\bigg) -\cos\bigg(\frac{\pi}{4}\bigg) = \frac{\sqrt{2}}{2} + \frac{\sqrt{2}}{2} = \sqrt{2}.$$ Now plug in $x= -\frac{\pi}{4}$: $$\sin\bigg(-\frac{\pi}{4}\bigg) -\cos\bigg(-\frac{\pi}{4}\bigg) = -\frac{\sqrt{2}}{2} - \frac{\sqrt{2}}{2} = -\sqrt{2}.$$

Thus, we have found that $\sqrt{2}$ is a maxiumum of $\sin(x) -\cos(x)$ and $-\sqrt{2}$ is a minimum. That is to say $$ -\sqrt{2} \leq \sin(x) -\cos(x) \leq \sqrt{2},$$ for all $x$. But by the definition of the absolute value, this means that $$ |\sin(x) -\cos(x)| \leq \sqrt{2},$$ and $\sqrt{2} < 2$, so we have $$ |\sin(x) -\cos(x)| \leq \sqrt{2} < 2.$$ And then we are done.

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The mean value theorem also works, since $$ \sin(x+\pi/2)-\sin x=-\frac{\pi}{2}\cos\xi, $$ where $\xi\in(x,x+\pi/2)$. It follows that $|\cos x-\sin x|\le \pi/2$. [This is, of course, not as sharp as achille hui's approach.]

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We have better inequality Since $|\cos a|\le 1$ and $$ \sqrt 2 (\cos(x+\frac{\pi}{4})) =\sqrt 2 (\cos x\cos\frac\pi4-\sin x\sin\frac\pi4) =\sin x -\cos y .$$ we have,

$$\color{blue}{|\sin x -\cos y| =\sqrt 2|\cos(x+\frac{\pi}{4})|\le\sqrt2<2.}$$

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