The tangent identity expresses the tangent of an angle in terms of the cosine and sine of the same angle.
One way to see why this is true for all angles is to draw the point corresponding to the angle of and write down the circle definitions of the trigonometric functions sine, cosine, and tangent.
The key insight is that and in the definitions of sine and cosine can be solved for and substituted into the definition of the tangent function like this.
The circle definitions used here are covered in more detail on the trigonometric functions page, and the rest of the identities are listed on the trigonometric identities page.