SOLUTION: Verify the identity: {{{tan2(x)tan(x)+2=(tan2(x))/(tan(x))}}}

Algebra ->  Trigonometry-basics -> SOLUTION: Verify the identity: {{{tan2(x)tan(x)+2=(tan2(x))/(tan(x))}}}      Log On


   

Question 882793: Verify the identity:
tan2%28x%29tan%28x%29%2B2=%28tan2%28x%29%29%2F%28tan%28x%29%29
Answer by Leaf W.(135) About Me  (Show Source):
You can put this solution on YOUR website!
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. ;-)
---
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%282x%29+=+2tan%28x%29%2F%281-%28tan%28x%29%29%5E2%29):
tan%282x%29%2Atan%28x%29+%2B+2
%282tan%28x%29%2F%281-%28tan%28x%29%29%5E2%29%29%2Atan%28x%29+%2B+2
2%28tan%28x%29%29%5E2%2F%281-%28tan%28x%29%29%5E2%29+%2B+2
---
Now, create a common denominator with 2 and combine:
2%28tan%28x%29%29%5E2%2F%281-%28tan%28x%29%29%5E2%29+%2B+2


%282%28tan%28x%29%29%5E2+%2B+2+-+2%28tan%28x%29%29%5E2%29%2F%281+-+%28tan%28x%29%29%5E2%29
2%2F%281+-+%28tan%28x%29%29%5E2%29
---
Therefore, we now have 2%2F%281+-+%28tan%28x%29%29%5E2%29+=+%28tan%282x%29%29%2F%28tan%28x%29%29. Let us now work from the right side:
%28tan%282x%29%29%2F%28tan%28x%29%29
%282tan%28x%29%2F%281-%28tan%28x%29%29%5E2%29%29%2F%28tan%28x%29%29
2%2F%281+-+%28tan%28x%29%29%5E2%29
---
That is the same as the left side, so we are now left with the true statement 2%2F%281+-+%28tan%28x%29%29%5E2%29+=+2%2F%281+-+%28tan%28x%29%29%5E2%29 -- the identity is verified. I hope this helps! =D