3.1.3. \( \cos ^{2} 225^{\circ}-\sin ^{2} 225^{\circ} \)

Did you know that the angle \(225^{\circ}\) is commonly associated with the coordinates of the point \((-1/\sqrt{2}, -1/\sqrt{2})\) on the unit circle? This angle is situated in the third quadrant, where both sine and cosine values are negative. By applying the cosine and sine values, you’ll find that \( \cos^2 225^{\circ} - \sin^2 225^{\circ} \) simplifies beautifully to \(-1\). Interestingly, this expression is a variant of the double angle formula for cosine, which is \( \cos(2\theta) = \cos^2(\theta) - \sin^2(\theta) \). This means that if you use \( \theta = 225^{\circ} \), you are effectively calculating \( \cos(450^{\circ}) \), which also equals \(-1\). Mathematics likes to surprise us with its interconnected truths!