[tex](\tan a+ \cot a)(1-\cos a)(1+\cos a)(1+\cot ^2 a) = \\ \\ =(\tan a+ \cot a)(1-\cos^2 a)(1+\cot ^2 a) = \\ \\ = (\tan a+ \cot a)\sin^2 a(1+\cot ^2 a) = \\ \\ = \Big(\dfrac{\sin a}{\cos a}+\dfrac{\cos a}{\sin a}\Big)\sin^2 a\Big(1+\dfrac{\cos^2 a}{\sin ^2 a}\Big) = \\ \\ = \dfrac{\sin^2 a+\cos^2 a}{\sin a\cos a}\sin^2 a \dfrac{\sin^2 a+\cos^2 a}{\sin^2 a} = \dfrac{1}{\sin a \cos a}\cdot 1 = \\ \\ = \dfrac{2}{2\sin a \cos a} = \dfrac{2}{\sin (2x)}[/tex]
