__ 19. (1+cosTheta )(cscTheta -cotTheta )

To simplify the expression \((1 + \cos\Theta)(\csc\Theta - \cot\Theta)\), we can use trigonometric identities and algebraic manipulation.<br /><br />First, recall the definitions of the trigonometric functions involved:<br />- \(\csc\Theta = \frac{1}{\sin\Theta}\)<br />- \(\cot\Theta = \frac}{\sin\Theta}\)<br /><br />Now, substitute these into the expression:<br /><br />\[<br />(1 + \cos\Theta)\left(\frac{1}{\sin\Theta} - \frac{\cos\Theta}{\sin\Theta}\right)<br />\]<br /><br />Combine the terms inside the parentheses:<br /><br />\[<br />(1 + \cos\Theta) \left(\frac{1 - \cos\Theta}{\sin\Theta}\right)<br />\]<br /><br />Next, distribute \((1 + \cos\Theta)\) across the numerator:<br /><br />\[<br />\frac{(1 + \cos\Theta)(1 - \cos\Theta)}{\sin\Theta}<br />\]<br /><br />Notice that \((1 + \cos\Theta)(1 - \cos\Theta)\) is a difference of squares:<br /><br />\[<br />(1 + \cos\Theta)(1 - \cos\Theta) = 1 - \cos^2\Theta<br />\]<br /><br />Using the Pythagorean identity \(1 - \cos^2\Theta = \sin^2\Theta\), we get:<br /><br />\[<br />\frac{\sin^2\Theta}{\sin\Theta}<br />\]<br /><br />Simplify the fraction:<br /><br />\[<br />\sin\Theta<br />\]<br /><br />Thus, the simplified form of the expression \((1 + \cos\Theta)(\csc\Theta - \cot\Theta)\) is:<br /><br />\[<br />\boxed{\sin\Theta}<br />\]