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Is it true that $\frac{\cos(x)}{x}=\frac{\sin(x)}{x-\frac{\pi}{2}}$?

I was going through a paper and I faced this equality.
$\frac{\cos(x)}{x}=\frac{\sin(x)}{x-\frac{\pi}{2}}$
This was proved to be true by simply replacing $x$ by $kx$ and then dividing both the sides with $k$. This is fair but I was wondering if there is any neat way to prove it.

A:

Let $t=x-\frac{\pi}{2}$, so that $\frac{\cos x}{x} = \frac{\cos(x-\frac{\pi}{2})}{x-\frac{\pi}{2}} = \frac{\cos t}{t}$ and $\frac{\sin x}{x} = \frac{\sin(x-\frac{\pi}{2})}{x-\frac{\pi}{2}} = \frac{\sin t}{t}$. Then you can replace $t$ with $\frac{\pi}{2}-t$ and you have
$$
\frac{\cos(t)}{t} = \frac{\sin t}{t-\pi/2} = \frac{\sin t}{ -\pi/2 + t} = \frac{\cos t}{ -\pi/2 + t} = \frac{\cos t}{x} = \frac{\cos x}{x}.
$$

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Schools can now educate game developers in an area as in-depth and interesting as game design. This way students can learn the basics of game development, reach a very high level, and move on to more diverse areas of creation like sound effects, animation, etc. Not only do schools benefit from the knowledge of their students but game developers gain a new pool of highly educated and creative people they can get involved with.

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