How can I prove that |sin x|≤1?

Answer 1

That's consistent with the definition of sine as the ratio of side opposite to hypotenuse. As the hypotenuse will always be the longest side ,sine can never have an absolute value > 1.

Answer 2

For any x, sinx is the second coordinate of a point on the unit circle.

So, -1 ≤ sinx ≤ 1.

That is, l sinx l ≤ 1.

Note, cosx is the x-coordinate of a point on the unit circle. So, l cosx l ≤ 1 as well.