\[ x=3 y=2 \] 20 나nory 2023 Home wark Given \( 2 \operatorname{sen} 2 \sin x-1=0, x \in=\left[0 \cdot 90^{\circ}\right] \) Crialore in tamat fr (2) \( 1-\sin ^{2} x \) (2) \( 1-\cos ^{2} x \) (4) \( \sin ^{2} x \) (S) \( \cos ^{2} x \) (4) Cithoar winy o Golculans wholis the easuer \( \sin ^{2} x+\cos ^{2} x=? \)

Let's break down the trigonometric equation \( 2 \sin^2 x - 1 = 0 \). By rearranging it, we can find that \( \sin^2 x = \frac{1}{2} \), which means that \( \sin x = \pm \frac{\sqrt{2}}{2} \). This implies that \( x \) can take the values of \( 45^{\circ} \) and \( 135^{\circ} \) in the specified range of \( [0, 90^{\circ}] \), but since \( 135^{\circ} \) exceeds this interval, the valid solution is \( x = 45^{\circ} \). A fun fact about trigonometric identities is that \( \sin^2 x + \cos^2 x = 1 \), which is known as the Pythagorean identity! This means that no matter what angle you choose, squaring both the sine and cosine of that angle and adding them together will always give you 1! It's like a magical shortcut in trigonometry. To do some additional practice, you might explore how to derive the unit circle. Understanding the unit circle not only visually represents sine and cosine values for different angles but also lays down the groundwork for advanced topics in trigonometry. Look for engaging resources, like interactive apps, to help solidify this knowledge!