Why do linear systems show sinusoidal fidelity?
A somewhat visual complement to the other answers

You are talking about systems that are linear and time invariant.

Exponential functions have one peculiar property (and can be actually defined by it): doing a time translation results in the same function multiplied by a constant. So

$$ e^{t-t_0}=e^{-t_0}e^t$$



The red exponential could as well be the blue one divided by $e$ or moved 1 second to the right

In general, this also holds for complex exponentials

Can you picture in your mind the plot of an complex harmonic such as $x(t)=e^{j2\pi t}$?
If so, you will see it's like a spring: it rotates along the complex plane as time goes by.



Rotating that spring (multiplying by a complex number in the unit circle) is the same as translating it. You probably have come into this visual effect some time in your life



It's the principle of any standard screw too.

Assume we input this in a linear time-invariant system. You get an output $y$
Now, input a rotated version of this spring. Because of linearity, the output should be $y$ rotated by the same amount. 
But since a rotation is equivalent to a time-translation, and the system is time-invariant, the output also has to be $y$ time-translated by the same amount.
So, $y$ has to satisfy the same property as the input: rotating it has to be equivalent to a particular time translation. This only happens when the output is a multiple of the original spring.

How much translation? Well, it's directly proportional to the rotation just like it would happen with a spring. The tighter the loops of the spring (the faster it rotates), the less it time-translates for a certain rotation. The tighter the loops of a screw, the more rounds you have to make it do for it to fit completely. And, when half of the rounds are done, the screw will be half it's way in... The output has to satisfy the same relation so, the output spring $y$ rotates at the same frequency as the input. 

At last, a reminder

$$\cos(t)=\frac{e^{j t}+e^{-j t}}{2}$$

$$\sin(t)=\frac{e^{j t}-e^{-j t}}{2 j}$$

So, that thing that happens with exponentials actually doesn't need to happen with cosines and sines in the most general case. But if the system is also real it's a different story...

In general, by this same reasoning, any exponential is an "eigenfunction" (output is proportional to input) of linear time invariant systems. That's why for these systems Z-transforms and Laplace transforms are so useful