Question 307231
<pre>
Rules for trig functions of negative values:

The cosine and secant are the only even trig functions,
and the negative sign "disappears":

{{{cos(-theta) = cos(theta)}}}
{{{sec(-theta) = sec(theta)}}}

All the other trig functions are odd trig functions,
and the negative sign comes out in front:

{{{sin(-theta) = -sin(theta)}}}
{{{tan(-theta) = -tan(theta)}}}
{{{csc(-theta) = -csc(theta)}}}
{{{cot(-theta) = -cot(theta)}}}

csc(-x)-csc(-x)cos<sup>2</sup>(x) becomes

-csc(x)-[-csc(x)]cos<sup>2</sup>(x) 

-csc(x)+csc(x)cos<sup>2</sup>(x)

Factor out -csc(x)

-csc(x)[1-cos<sup>2</sup>(x)]

Us the identity {{{Sin^2theta+Cos^2theta=1}}} written as {{{Sin^2theta=1-Cos^2theta}}} to replace the bracketed expression:

-csc(x)sin<sup>2</sup>(x)

Use the identity {{{csc(theta)=1/sin(theta)}}} to replace the cosecant factor:

-{{{1/sin(x)}}}sin<sup>2</sup>(x)

This simplifies to

-sin(x)

Edwin</pre>