Draw a right triangle with angle x, hypotenuse 1, and legs "a" and "b"


 


Since , that's
because by the Pythagorean theorem, a2+b2=1



Multiply top and bottom by a2b2



We multiply out the denominator,



Since a2+b2=1,





The denominator factors



Multiply by the product of the "conjugates" of the
factors on the bottom over itself. (Sort of like
rationalizing the denominator)




 


By the Pythagorean theorem  1-a2=b2 and 1-b2=a2



Canceling:



Since a = sin(x) and b = cos(x)

  <--simplest form

Edwin