- #1

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If I'm given the Pythagorean theorem, a^2 + b^2 = c^2, and asked to prove it, then I say there is a triangle, with hypotenuse, c, the horizontal leg of the triangle, a, and the vertical leg of the triangle, b.

Then CosX = a/c and SinX = b/c,

Therefore a = cCosX, b = cSinX and c = a/CosX

I can then substitute those into the theorem a^2 + b^2 = c^2 and get:

a^2/(CosX)^2 = (c^2)(CosX)^2 + (c^2)(SinX)^2

Simplifying:

a^2/(CosX)^2 = (c^2)(1 - (SinX)^2) + (c^2)(SinX)^2

a^2/(CosX)^2 = c^2 - (c^2)(SinX)^2 + (c^2)(SinX)^2

a^2/(CosX)^2 = c^2

Taking the square root of both sides:

a/CosX = c

Which is what I said c was equal to in the beginning. Basically, I assumed the premise, and since I arrived at a truth, I assumed the theorem is true.

I am pretty sure that the Cosine and Sine ratios are proved using Pythagoren theorem, so that automatically invalidates my proof, but is the method (assuming the premise then arriving at another basic truth) valid?