We are given the main identity of cos^2+sin^2=1. This identity is derived from the ratios of cos and sine in a right triangle. Cos is equal to x/1, that will be the first part of the equation. Sin is equal to y/1, that will be the second part of the equation. R=1. So our equation of the pythagoreon is x^2+y^2=r^2. By substituting the ratios we get cos^2+sin^2=1.
When we divide cos^2 to the entire eqution, we get the dervation of the next Pythagoreon identity: 1+tan^2=sec^2. *This occurs because cos/cos=1. We know that sin/cos=tan through the ratio identities. And finally, 1/cos=sec.
Our next derivation is derived when we dived sin to the entire equation: cot^2+1=csc^2. *Dividing sin will do this: cos/sin=cot, which is what we know from the ratio identity. sin/sin=1. And finally, 1/sin=csc.
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