Question 1154341
<pre>

{{{tan^2(x)(1+cot^2(x)^"")=1/(1-sin^2(x))}}}

Work with the left side:

{{{tan^2(x)(1+cot^2(x)^"")}}}

Distribute:

{{{tan^2(x)+tan^2(x)cot^2(x)^"")}}}

Change the left side to sines and cosines:

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

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

Cancel:

{{{(sin^2(x)^""/cos^2(x)^"")+(cross(sin^2(x)^"")/cross(cos^2(x)^""))(cross(cos^2(x)^"")/cross(sin^2(x)^""))}}}

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

Get LCD:

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

Combine numerators over LCD:

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

Use Pythagorean identity in numerator:

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

Use Pythagorean identity in denominator:

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

Edwin</pre>