Answer:
[tex]\dfrac{1+\cos(x)}{\sin(x)}+\dfrac{\sin(x)}{1+\cos(x)}[/tex]
[tex]=\dfrac{(1+\cos(x))(1+\cos(x))}{\sin(x)(1+\cos(x))}+\dfrac{\sin^2(x)}{\sin(x)(1+\cos(x))}[/tex]
[tex]=\dfrac{1+2 \cos(x)+\cos^2(x)+\sin^2(x)}{\sin(x)(1+\cos(x))}[/tex]
[tex]=\dfrac{1+2 \cos(x)+1}{\sin(x)(1+\cos(x))}[/tex]
[tex]=\dfrac{2+2 \cos(x)}{\sin(x)(1+\cos(x))}[/tex]
[tex]=\dfrac{2(1+ \cos(x))}{\sin(x)(1+\cos(x))}[/tex]
[tex]=\dfrac{2}{\sin(x)}[/tex]
[tex]=2 \csc(x)[/tex]
