How to derive pythagorean identity?

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To derive the Pythagorean identity sin²(x) + cos²(x) = 1, one approach involves using the angle addition formula for cosine, specifically setting y such that x + y = 0, which simplifies to cos(x + y) = 1. Understanding that cosine is an even function and sine is an odd function is crucial in this derivation. A more intuitive explanation connects the identity to the unit circle, where the coordinates correspond to cos(x) and sin(x), illustrating the relationship to Pythagorean theorem. Some participants express confusion about the choice of y over x and the relevance of even and odd functions in the derivation. Overall, visualizing the identity through the unit circle can clarify its geometric significance.
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I got some precalc review to prepare for calc, and after hours of doing the packet, I'm on the last problem set...but it's all about derivatives which we never touched on last year.

Homework Statement


I'm supposed to derive sin^2 + cos^2 = 1


Homework Equations


It says to use cos 0 =1, cos (x+y) = cos x cos y - sin x sin y, but I have no idea how to use these.


please help! I'm absolutely clueless at math :frown:
 
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All you need to do is choose y so that x+y = 0 and thus cos (x+y) = 1 and plug your values into the right hand side of the equation you're given. You will also need to know about cos being an even function and sin being an odd function.
 
Thanks! I got it
 
Its not the best derivation of the result because it leaves you wondering what Pythagoras has to do with it. Its best to derive this result from the unit circle where the x coordinate is given by cos(x) and the y coordinate by sin (x) then it becomes immediately apparent where Pythagoras comes in.
 
That interseting because i just derived pythagoras c^2=a^2+b^2 formula from an ellipse, very enlightening.
 
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sorry but can anyone please explain it to me again? i don't understand why you have to choose a value for y and not x and i don't see how even/odd functions will be used
 

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