Why would knowing the Pythagorean identity sin2x + cos2x = 1 and the sum identities allow you to prove all the other identities?

If you divided each side by (sinx) (sinx), the result is 1 + (cotx) (cotx) = (cscx) (cscx)

If you divide each side by (cosx) (cosx), the resut is 1 + (tanx) (tanx) = (secx) (secx)

These are the basic trigonometric identities.

These are simple trigonometric identities.