# Prove this

• Jul 23rd 2012, 08:16 AM
kjchauhan
Prove this

$\displaystyle tan^{-1}\left(\frac{tan2\theta + tanh2\phi}{tan2\theta - tanh2\phi}\right)+tan^{-1}\left(\frac{tan\theta - tanh\phi}{tan\theta + tanh\phi}\right)=tan^{-1}\left(cot\theta coth\phi \right)$

Thank You Very Much....
• Jul 23rd 2012, 12:53 PM
ebaines
Re: Prove this
Are you sure you typed the formula correctly? I don't believe the two sides are equal. I tried values of theta = 0.1 and phi = 0.2 and I get a result of -1.59 for the left side and +1.55 for the right side.
• Jul 23rd 2012, 04:31 PM
kjchauhan
Re: Prove this
Yes, I typed correctly as given in the book, may be it was printed wrong in book, thanks ..
• Jul 23rd 2012, 06:50 PM
Amer
Re: Prove this
Quote:

Originally Posted by kjchauhan

$\displaystyle tan^{-1}\left(\frac{tan2\theta + tanh2\phi}{tan2\theta - tanh2\phi}\right)+tan^{-1}\left(\frac{tan\theta - tanh\phi}{tan\theta + tanh\phi}\right)=tan^{-1}\left(cot\theta coth\phi \right)$

Thank You Very Much....

take the tan for both sides
left side
$\displaystyle tan \left(tan^{-1}\left(\frac{tan2\theta + tanh2\phi}{tan2\theta - tanh2\phi}\right)+tan^{-1}\left(\frac{tan\theta - tanh\phi}{tan\theta + tanh\phi}\right)\right)$

then use the identity $\displaystyle \tan (a+b) = \frac{\tan a + \tan b }{1 - \tan a \tan b }$
resulting

$\displaystyle \dfrac{\dfrac{tan2\theta + tanh2\phi}{tan2\theta - tanh2\phi} + \dfrac{tan\theta - tanh\phi}{tan\theta + tanh\phi}}{1 - \left(\dfrac{tan2\theta + tanh2\phi}{tan2\theta - tanh2\phi}\right)\left(\dfrac{tan\theta - tanh\phi}{tan\theta + tanh\phi}\right)}$

we have to prove that the previous equal $\displaystyle \cot \theta \coth \phi$

You can make the nominator and denominator with common denominator then cancel it
• Jul 23rd 2012, 11:38 PM
kjchauhan
Re: Prove this
Thank you very much for giving such an important hint..
• Jul 24th 2012, 05:08 AM
ebaines
Re: Prove this
Quote:

Originally Posted by ebaines
Are you sure you typed the formula correctly? I don't believe the two sides are equal. I tried values of theta = 0.1 and phi = 0.2 and I get a result of -1.59 for the left side and +1.55 for the right side.

Update - the formula is indeed correct. My mistake was in relying on my PC to provide the arctan calculation, and forgetting that it can be plus or minus pi. With that correction the formula does indeed work. Sorry for the earlier post.