What is a mathematical identity?

Question

Why are the three Pythagorian identities called Pythagorian identities? How do you prove them?

Answer 1

And identity is a statement which will be always be true.
For instance

sin² Θ + cos² Θ = 1.

No matter what we put in for Θ, the square of its sine added to the square of its cosine will yield a sum of 1.

These are proven using other theorems, postulates and definitions.

Answer 2

An identity is an equality that is always true for any values of the variables intervening in it.
The first of the pythagorian identities:
sin^2(x)+ cos^2(x)=1 is always true for any x because it expresses the Pythagoras theorem (for a right triangle), that has multiple demostration procedures.
The same is true for the other two pythagorian identities, that are different forms of the same theorem, with other trigonometric functions, sec, tan, cot, and cosec.

Answer 3

The pythagoream theorem is c^2 = a^2 +b^2 which was proved by Euclid over 2000 years ago. His proof is elegant.

The three Pythagorean identities are based on this theorem:

cos^2x +sin^2x = 1 <-- Identity #1

1 + tan^2x = sec^2x <-- Identity #2

cot^2x +1 = csc^2x <-- Identity #3

Identity #1 follows directly from the Pythagorean Theorem, since in a unit circle (radius =1) x = sin z and y = coz and x^2+y^2 = 1^2
So sin^2z + cos^2z =1

Identity #2 is obtained by dividing Identity #1 by cos^2x

Identity #3 is obtained by multiplying Identity #2 by cot^2x

Answer 4

you dotn have to since they are already proven by some one else