Pythagorean identities

Proof of the Pythagorean identities

Proof 1.   According to the Pythagorean theorem,

x² + y² = r².  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .(1)

Therefore, on dividing both sides by r²,

cos²θ  +  sin²θ  =  1.  .  .  .  .  .  .  .  .  .  .  .  .  .(2)

Apart from the order of the terms, this is the first Pythagorean identity, a)

To derive b), divide line (1) by x²; to derive c), divide by y².

Or, we can derive both b) and c) from a) by dividing it first by cos²θ and then by sin²θ.  On dividing line 2) by cos²θ, we have

That is,

1 + tan²θ  =  sec²θ.

And if we divide a) by sin²θ, we have

That is,

1 + cot²θ  =  csc²θ.

The 3 Pythagorean identities are thus equivalent to one another.

Proof 2.