Question 425709
Prove the identity:

{{{-Cotx + Sinx/(1-Cosx) =Cscx}}}
<pre><font face = "consolas" color = "indigo" size = 2><b>

We will work with just the left side:

Use the quotient identity {{{Cot(theta)=Cos(theta)/Sin(theta)}}}
to rewrite the first term:

{{{-(Cosx)/(Sinx) + Sinx/(1-Cosx)}}}


Multiply the second fraction by {{{red(((1+Cosx))/((1+Cosx)))}}}
which just equals 1:


{{{-(Cosx)/(Sinx) + expr(Sinx/((1-Cosx)))expr(red(((1+Cosx))/((1+Cosx))))}}} 

Multiply the fractions:

{{{-(Cosx)/(Sinx) + (Sinx(1+Cosx))/(1-Cos^2x)}}}

Use the Pythagorean identity {{{Sin^2theta+Cos^2theta=1}}} written as {{{1-Cos^2theta=Sin^2theta}}}
to replace the second denominator:

{{{-(Cosx)/(Sinx) + (Sinx(1+Cosx))/(Sin^2x)}}}

Simplify the second term on the bottom:

{{{-(Cosx)/(Sinx) + (cross(Sinx)(1+Cosx))/(Sin^cross(2)x)}}}

{{{-(Cosx)/(Sinx) + (1+Cosx)/(Sinx)}}}

The denominators are the same so we can combine the
two fractions by combining the numerators over the
common denominator:

{{{(-Cosx+1+Cosx)/(Sinx)}}}

Simplify the numerator:

{{{(cross(-Cosx)+1+cross(Cosx))/(Sinx)}}}

{{{1/(Sinx)}}}

Use the reciprocal identity {{{Csc(theta)=1/Sin(theta)}}}

{{{Csc(x)}}}

Edwin</pre>