I learned that sin^2(theta) + cos^2(theta) = 1, but for a problem we did, sin^2(15) - cos^2(15) = 1. I don't understand how this works, how do you use the Pythagorean Identity to manipulate the steps into getting 1?
Use the identity $\displaystyle \cos^2\theta-\sin^2\theta=\cos2\theta.$ In your example, $\displaystyle \sin^215^\circ-\cos^215^\circ = -\cos30^\circ = -\frac{\sqrt3}2.$
Wait, so then we solved the problem incorrectly? the answer is not equal to 1? Ugh...I wish I knew when a solution was wrong...I spent all afternoon trying to decipher how the arrived was arrived at.
Wait, so then we solved the problem incorrectly? the answer is not equal to 1? Ugh...I wish I knew when a solution was wrong...I spent all afternoon trying to decipher how the arrived was arrived at.
yes ... you evaluated $\displaystyle \sin^2(15^\circ) - \cos^2(15^\circ)$ incorrectly