Question 167728
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Verify this identity : 
(tan^2(x)-1)/(1+tan^2(x)) = 1-2cos^2(x) 
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~


It is done in 3 (three) easy steps.


<pre>
1.  The numerator tan^2(x) - 1 = {{{sin^2(x)/cos^2(x) -1}}} = {{{(sin^2(x)-cos^2(x))/cos^2(x)}}}.                 (1)


2.  The denominator 1 + tan^2(x) = {{{1 + sin^2(x)/cos^2(x)}}} = {{{(sin^2(x) + cos^2(x))/cos^2(x)}}} = {{{1/cos^2(x)}}}.      (2)


3.  When you divide the numerator (expression (1)) by the denominator (expression (2)), cos^2(x) cancels, 
    and the remaining expression is  {{{sin^2(x)-cos^2(x)}}},  which you can rewrite

    {{{sin^2(x)-cos^2(x)}}} = {{{(1-cos^2(x)) - cos^2(x)}}} = {{{1 - 2*cos^2(x)}}}.


    It is precisely your right side.
</pre>


For other similar solved problems see the lesson

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/Trigonometry-basics/Proving-Trigonometry-identities.lesson>Proving Trigonometry identities</A>

in this site.



Also, you have this free of charge online textbook in ALGEBRA-II in this site

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/complex/ALGEBRA-II-YOUR-ONLINE-TEXTBOOK.lesson>ALGEBRA-II - YOUR ONLINE TEXTBOOK</A>.


The referred lesson is the part of this online textbook under the topic "<U>Trigonometry: Solved problems</U>".



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