__ 11 (cscx)/(sinx)-(cotx)/(tanx)

To simplify the expression \(\frac{\csc x}{\sin x} - \frac{\cot x}{\tan x}\), we need to use trigonometric identities and properties.<br /><br />First, recall the following trigonometric identities:<br />1. \(\csc x = \frac{1}{\sin x}\)<br />2. \(\cot x = \frac{\cos x}{\sin x}\)<br />3. \(\tan x = \frac{\sin x}{\cos x}\)<br /><br />Now, let's rewrite the given expression using these identities:<br /><br />\[<br />\frac{\csc x}{\sin x} - \frac{\cot x}{\tan x} = \frac{\frac{1}{\sin x}}{\sin x} - \frac{\frac{\cos x}{\sin x}}{\frac{\sin x}{\cos x}}<br />\]<br /><br />Simplify each term separately:<br /><br />1. \(\frac{\frac{1}{\sin x}}{\sin x} = \frac{1}{\sin x \cdot \sin x} = \frac{1}{\sin^2 x} = \csc^2 x\)<br />2. \(\frac{\frac{\cos x}{\sin x}}{\frac{\sin x}{\cos x}} = \frac{\cos x}{\sin x} \cdot \frac{\cos x}{\sin x} = \frac{\cos^2 x}{\sin^2 x} = \cot^2 x\)<br /><br />So the expression becomes:<br /><br />\[<br />\csc^2 x - \cot^2 x<br />\]<br /><br />Next, use the Pythagorean identity for cosecant and cotangent:<br /><br />\[<br />\csc^2 x = 1 + \cot^2 x<br />\]<br /><br />Substitute this identity into the expression:<br /><br />\[<br />\csc^2 x - \cot^2 x = (1 + \cot^2 x) - \cot^2 x = 1<br />\]<br /><br />Therefore, the simplified form of the given expression is:<br /><br />\[<br />\boxed{1}<br />\]