\[ x=3 y=2 \] 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 problem step by step. ### Given Equation We have the equation: \[ 2 \sin 2x - 1 = 0 \] ### Step 1: Solve for \( \sin 2x \) First, we can isolate \( \sin 2x \): \[ 2 \sin 2x = 1 \] \[ \sin 2x = \frac{1}{2} \] ### Step 2: Find \( 2x \) Next, we need to find the angles where \( \sin 2x = \frac{1}{2} \). The general solutions for \( \sin \theta = \frac{1}{2} \) are: \[ \theta = 30^\circ + k \cdot 360^\circ \quad \text{and} \quad \theta = 150^\circ + k \cdot 360^\circ \] for any integer \( k \). ### Step 3: Solve for \( x \) Since \( 2x = 30^\circ \) or \( 2x = 150^\circ \), we can divide by 2 to find \( x \): 1. \( 2x = 30^\circ \) gives \( x = 15^\circ \) 2. \( 2x = 150^\circ \) gives \( x = 75^\circ \) ### Step 4: Check the interval We need to ensure that \( x \) is within the interval \( [0, 90^\circ] \). Both \( 15^\circ \) and \( 75^\circ \) are valid solutions. ### Step 5: Evaluate \( \sin^2 x + \cos^2 x \) Now, we need to calculate \( \sin^2 x + \cos^2 x \). According to the Pythagorean identity: \[ \sin^2 x + \cos^2 x = 1 \] ### Summary of Results 1. The solutions for \( x \) are \( 15^\circ \) and \( 75^\circ \). 2. The value of \( \sin^2 x + \cos^2 x \) is: \[ \sin^2 x + \cos^2 x = 1 \] Thus, the final answer is: \[ \sin^2 x + \cos^2 x = 1 \]