Question 882793
Wow! Congratulations; you are one of the very few people (maybe the first person I have ever seen!) to actually use the formula notation! Thank you; it makes it easier for us that way. ;-)
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Let us first work from the left side and see if we can get to the right side. First, use the formula for tan(2x) ({{{tan(2x) = 2tan(x)/(1-(tan(x))^2)}}}):
{{{tan(2x)*tan(x) + 2}}}
{{{(2tan(x)/(1-(tan(x))^2))*tan(x) + 2}}}
{{{2(tan(x))^2/(1-(tan(x))^2) + 2}}}
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Now, create a common denominator with 2 and combine:
{{{2(tan(x))^2/(1-(tan(x))^2) + 2}}}
{{{2(tan(x))^2/(1-(tan(x))^2) + 2(1 - (tan(x))^2)/(1 - (tan(x))^2)}}}
{{{2(tan(x))^2/(1-(tan(x))^2) + (2 - 2(tan(x))^2)/(1 - (tan(x))^2)}}}
{{{(2(tan(x))^2 + 2 - 2(tan(x))^2)/(1 - (tan(x))^2)}}}
{{{2/(1 - (tan(x))^2)}}}
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Therefore, we now have {{{2/(1 - (tan(x))^2) = (tan(2x))/(tan(x))}}}. Let us now work from the right side:
{{{(tan(2x))/(tan(x))}}}
{{{(2tan(x)/(1-(tan(x))^2))/(tan(x))}}}
{{{2/(1 - (tan(x))^2)}}}
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That is the same as the left side, so we are now left with the true statement {{{2/(1 - (tan(x))^2) = 2/(1 - (tan(x))^2)}}} -- the identity is verified. I hope this helps! =D