The Pythagorean Theorem gives the relationship between the sides of a right triangle. The following identities show the relationships between the trigonometric functions of a particular angle.
sin^2 θ + cos^2 θ = 1
tan^2 θ + 1 = sec^2 θ
1 + cot^2 θ = csc^2 θ

Why then are these referred to as the Pythagorean Identities?



Is the answer to this question:
because they are showing and proving the Pythadorean theorum through trigonometry? I'm not sure how else to answer this...

They show the basic relations between the sine and cosine functions, from which all others may be derived.

Just to add an example: if the two legs of a right angle triangle are A and B and the hypotenuse is C, then Pythagorus says that A^2 + B^2 = C^2. Now consider as an example the second equation you wrote: tan^2 \theta + 1 = sec^2 \theta :


tan^2 \theta + 1 = \frac {A^2}{B^2} + 1 = \frac {A^2 + B^2}{B^2} = \frac {C^2}{B^2} = sec^2 \theta