Question 1012353
<pre>
{{{(cot^2(x) - tan^2(x)) / (cot^""(x)^"" + tan^""(x))^2}}}{{{""=""}}}{{{2cos^2(x)-1}}}

Work with the left side. Factor the numerator as the difference
of two squares.
Write the denominator as the product of 
two equal factors:

{{{((cot^""(x)^"" - tan^""(x))(cot^""(x)^"" + tan^""(x))) / ((cot^""(x)^"" + tan^""(x))(cot^""(x)^"" + tan^""(x)))}}}
Cancel:

{{{((cot^""(x)^"" - tan^""(x))cross((cot^""(x)^"" + tan^""(x)))) / ((cot^""(x)^"" + tan^""(x))(cross(cot^""(x)^"" + tan^""(x))))}}}

{{{(cot^""(x) - tan^""(x)) / (cot^""(x) + tan^""(x))}}}

Use the quotient identities to change everything to sines
and cosines:

{{{(cos(x)/sin(x) - sin(x)/cos(x)) / (cos(x)/sin(x) + sin(x)/cos(x))}}}

Multiply top and bottom by LCD = sin(x)cos(x)

{{{( cos^2(x)-sin^2(x) )/(cos^2(x)+sin^2(x))}}}

Use the Pythagorean identity to replace the bottom by 1:

{{{(cos^2(x)-sin^2(x))/1}}}

{{{cos^2(x)-sin^2(x)}}}

Use the Pythagorean identity to change the sin<sup>2</sup>(x)

{{{cos^2(x)-(1-cos^2(x)^"")}}}

{{{cos^2(x)-1+cos^2(x)}}}

{{{2cos^2(x)-1}}}

Edwin</pre>