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

To simplify the expression \((1 + \cos\Theta)(\csc\Theta - \cot\Theta)\), we can use trigonometric identities and algebraic manipulation.First, recall the definitions of the trigonometric functions involved:- - Now, substitute these into the expression: Combine the terms inside the parentheses: Next, distribute \((1 + \cos\Theta)\) across the numerator: Notice that \((1 + \cos\Theta)(1 - \cos\Theta)\) is a difference of squares: Using the Pythagorean identity , we get: Simplify the fraction: Thus, the simplified form of the expression \((1 + \cos\Theta)(\csc\Theta - \cot\Theta)\) is: