QUESTION 5 DO NOT USE A CALCULATOR FOR THIS QUESTION. 5.1 Complete the following identities: 5.1.1 \( \quad \cos ^{2} \mathrm{~A}+\sin ^{2} \mathrm{~A}=\ldots \) \( 5.1 .2 \quad \cos ^{2} \mathrm{~A}-\sin ^{2} \mathrm{~A}=\ldots \)

The first identity, \( \cos^{2} A + \sin^{2} A = 1 \), is a cornerstone of trigonometry and stems from the Pythagorean theorem. Imagine the unit circle where every point on the circle corresponds to an angle \( A \); the coordinates of those points are given by \( (\cos A, \sin A) \). When you add the squares of these coordinates, you're essentially finding the radius squared, which is always 1! The second identity, \( \cos^{2} A - \sin^{2} A = \cos(2A) \), showcases the beauty of double angle formulas. It’s like a magic trick in trigonometry! By combining the squares of cosine and sine into one elegant expression, we see how the behavior of angles doubles up — perfect for simplifying calculations or deriving more complex identities, especially in problems involving periodic functions.