Question

Please answer the 4th

Answer 1

$\: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: your \: \: answer \\ \\ \\ \tan( \frac{ \cos(x) }{1 - \sin(x) } ) - \cot( \sqrt{ \frac{1 + \cos(x) }{1 - \cos(x) } } ) \\ \\ we \: \: know \: \: that \\ \\ \cos(x) = ( \cos( \frac{x}{2} ) + \sin( \frac{x}{2} ) )( \cos( \frac{x}{2} ) - \sin( \frac{x}{2} ) ) \\ 1 - \sin(x) = ( \cos( \frac{x}{2} ) - \sin( \frac{x}{2} ) ) {}^{2} \\ 1 + \cos(x) = 2 \cos {}^{2} ( \frac{x}{2} ) \\ an d\\ 1 - \cos(x) = 2 \sin {}^{2} ( \frac{ x }{2} ) \\ \\ \\ \tan( \tan {}^{ - 1} ( \frac{\pi}{4} + \frac{x}{2} ) ) - \cot( \cot {}^{ - 1} ( \frac{x}{2} ) ) \\ \\ \frac{\pi}{4} + \frac{x}{2} - \frac{x}{2} \\ \\ = \frac{\pi}{4}$