xin (0,(pi )/(2)) (cos(frac (3pi )/(2)-x)-sin(2pi -x))(tan(pi +x))=?

To solve this expression, let's simplify each trigonometric function step by step:<br /><br />1. $\cos\left(\frac{3\pi}{2} - x\right)$:<br /> - This is the cosine of an angle in the second quadrant.<br /> - We can use the identity $\cos(\pi - \theta) = -\cos(\theta)$.<br /> - Applying this identity, we get $\cos\left(\frac{3\pi}{2} - x\right) = -\cos(x)$.<br /><br />2. $\sin(2\pi - x)$:<br /> - This is the sine of an angle in the fourth quadrant.<br /> - We can use the identity $\sin(2\pi - \theta) = -\sin(\theta)$.<br /> - Applying this identity, we get $\sin(2\pi - x) = -\sin(x)$.<br /><br />3. $\tan(\pi + x)$:<br /> - This is the tangent of an angle in the third quadrant.<br /> - We can use the identity $\tan(\pi + \theta) = \tan(\theta)$.<br /> - Applying this identity, we get $\tan(\pi + x) = \tan(x)$.<br /><br />Now, let's substitute these simplified expressions back into the original expression:<br /><br />$\frac{\cos\left(\frac{3\pi}{2} - x\right) - \sin(2\pi - x)}{\tan(\pi + x)} = \frac{-\cos(x) - (-\sin(x))}{\tan(x)}$<br /><br />Simplifying further:<br /><br />$\frac{-\cos(x) + \sin(x)}{\tan(x)}$<br /><br />We know that $\tan(x) = \frac{\sin(x)}{\cos(x)}$, so we can substitute this into the expression:<br /><br />$\frac{-\cos(x) + \sin(x)}{\frac{\sin(x)}{\cos(x)}}$<br /><br />Simplifying the fraction:<br /><br />$\frac{-\cos(x) + \sin(x)}{\frac{\sin(x)}{\cos(x)}} = \frac{(-\cos(x) + \sin(x)) \cdot \cos(x)}{\sin(x)}$<br /><br />Expanding the numerator:<br /><br />$\frac{-\cos^2(x) + \sin(x) \cdot \cos(x)}{\sin(x)}$<br /><br />Now, we can simplify the expression by canceling out common factors:<br /><br />$\frac{-\cos^2(x) + \sin(x) \cdot \cos(x)}{\sin(x)} = \frac{\cos(x)(\sin(x) - \cos(x))}{\sin(x)}$<br /><br />Since $x \in (0, \frac{\pi}{2})$, we know that $\sin(x) > 0$. Therefore, we can simplify further:<br /><br />$\frac{\cos(x)(\sin(x) - \cos(x))}{\sin(x)} = \cos(x) - \frac{\cos^2(x)}{\sin(x)}$<br /><br />So, the simplified expression is:<br /><br />$\cos(x) - \frac{\cos^2(x)}{\sin(x)}$<br /><br />This is the final simplified form of the given trigonometric expression.