Question 156515
Simplify:
{{{((csc(x)-cot(x))(csc(x)+cot(x)))/csc(x)}}}
<pre><font size = 4 color="indigo"><b>
FOIL out the top:

{{{(csc^2x+csc(x)cot(x)-cot(x)csc(x)-cot^2x)/csc(x)}}}

{{{(csc^2x+cross(csc(x)cot(x)-cot(x)csc(x))-cot^2x)/csc(x)}}}

{{{(csc^2x-cot^2x)/csc(x)}}}

use the Pythagorean identity {{{1+cot^2alpha=csc^2alpha}}} which
can be rearranged as {{{csc^2alpha-cot^2alpha=1}}} to replace
the numerator by just {{{1}}}.

{{{1/csc(x)}}}

Then use the reciprocal identity {{{csc(alpha)=1/sin(alpha)}}}
to replace the denominator:

{{{1/(1/sin(x))}}}

{{{1}}}÷{{{1/(sin(x))}}}

Invert the second fraction and change division to
multiplication:

{{{1}}}×{{{(sin(x))/1}}}

{{{sin(x)}}}

Edwin</pre>