When the hypotenuse equals 1.

We know that the trigonometric functions are defined as ratios in a right triangle as follows:

sine (sin) = opposite/hypotenuse

cosine (cos) = adjacent/hypotenuse

tangent (tan) = opposite/adjacent

Consider all the similar right triangles that include the acute angle we call alpha in this picture.  If we think about the triangle where the hypotenuse = 1, our formulas get much simpler.

sin alpha = opposite

cos alpha = adjacent

tan alpha = opposite/adjacent = sin alpha/cos alpha

The tangent of alpha ALWAYS equals sin alpha/cos alpha, but to make sine and cosine equal to the lengths of sides of a triangle, we need the hypotenuse equal to 1.  Here are some other true statements given our picture.

1. sin alpha is the height of this triangle.
2. cos alpha is the base of the triangle, here labeled x.
3. If we have the triangle with the perpendicular lines at the horizontal and vertical and the angle alpha is on the left side, tan alpha is the slope of the hypotenuse.

And here is a statement that is always true for any angle alpha, a statement known as The Trigonometric Identity.

sin² alpha + cos² alpha = 1.

This is a very straightforward application of the Pythagorean Theorem, since all it is says is that

a²/c² + b²/c² = 1.

This is true because when we combine the two fractions, we get

(a² + b²)/c², which is the same as c²/c² = 1.

I say this statement is The Trigonometric Identity.  Other sources will say there are several trigonometric identities.  There are others, but we will see that all of them are just rearrangements of this one.  Make sure you understand this.  Quite often when you do not know the next step in a trig problem, The Trigonometric Identity is the correct path to take.