SOLUTION: I need to write the following as an algebraic expression: sin(2arccos x)

let 

then 

So draw a right triangle and label one of the acute angles 


 

Now we need to put values on the sides of the
triangle to cause  to equal .

Since the cosine is the adjacent side divided by the
hypotenuse, if we label the adjacent side as x and the
hypotenuse as 1. then  will equal .



Now we use the Pythagorean theorem to find the opposite
side:











So we write this on the opposite side:



Now we go back to the original problem:



or



and we use the identity

  

Then using the fact that the sine is the opposite over
the hypotenuse and the cosine is the adjacent over the
hypotenuse, we use the right triangle above:









Edwin