Question

Mostre que cotgx + cossecx / senx = 1 / 1- cosx

Answer 1

Note que:
cotgx= $\frac{1}{tgx} =$$\frac{cosx}{senx}$;
cossecx = $\frac{1}{senx}$;
$sen^{2}x+cos^{2}x=1$.

Então:
$\frac{cotgx+cossecx}{senx} = \\ \frac{ \frac{cosx}{senx}+ \frac{1}{senx} }{senx} = \\ \frac{ \frac{(cosx+1)}{senx} }{senx} = \\ \frac{cosx+1}{sen^{2}x} = \\ \frac{cosx+1}{1-cos^{2}x} = \\ \frac{cosx+1}{(1-cosx)(1+cosx)} = \\ \frac{1}{1-cosx} =$

c.q.d.