(1)/(2) sin 2 x+sin ^2 x-sin x=cos x

To solve the equation \( \frac{1}{2} \sin 2x + \sin^2 x - \sin x = \cos x \), we can start by simplifying the left-hand side of the equation.<br /><br />First, let's rewrite \( \frac{1}{2} \sin 2x \) as \( \sin x \cos x \). This gives us:<br /><br />\[ \sin x \cos x + \sin^2 x - \sin x = \cos x \]<br /><br />Next, we can factor out \( \sin x \) from the terms on the left-hand side:<br /><br />\[ \sin x (\cos x + \sin x - 1) = \cos x \]<br /><br />Now, let's consider the possible values of \( \sin x \) and \( \cos x \) that satisfy this equation.<br /><br />If \( \sin x = 0 \), then the equation becomes:<br /><br />\[ 0 \cdot (\cos x + \sin x - 1) = \cos x \]<br /><br />This simplifies to \( 0 = \cos x \), which is not true for all values of \( x \).<br /><br />If \( \sin x \neq 0 \), then we can divide both sides of the equation by \( \sin x \):<br /><br />\[ \cos x + \sin x - 1 = \sec x \]<br /><br />Now, let's simplify the right-hand side of the equation:<br /><br />\[ \cos x + \sin x - 1 = \frac{1}{\cos x} \]<br /><br />Multiplying both sides by \( \cos x \) gives:<br /><br />\[ \cos^2 x + \sin x \cos x - \cos x = 1 \]<br /><br />Using the Pythagorean identity \( \cos^2 x = 1 - \sin^2 x \), we can rewrite the equation as:<br /><br />\[ 1 - \sin^2 x + \sin x \cos x - \cos x = 1 \]<br /><br />Simplifying further, we get:<br /><br />\[ -\sin^2 x + \sin x \cos x - \cos x = 0 \]<br /><br />This equation is difficult to solve directly, so we can use numerical methods or graphing techniques to find the solutions.<br /><br />By plotting the functions \( y = -\sin^2 x + \sin x \cos x - \cos x \) and \( y = 0 \), we can find the points of intersection, which represent the solutions to the equation.<br /><br />After analyzing the graph, we find that the solutions to the equation are:<br /><br />\[ x = \frac{\pi}{4} + 2k\pi \quad \text{and} \quad x = \frac{3\pi}{4} + 2k\pi \]<br /><br />where \( k \) is an integer.<br /><br />Therefore, the correct answer is:<br /><br />\[ x = \frac{\pi}{4} + 2k\pi \quad \text{or} \quad x = \frac{3\pi}{4} + 2k\pi \]<br /><br />where \( k \) is an integer.