Prove this

**kjchauhan** · July 23rd 2012, 08:16 AM

Please Help me to prove this,

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

Thank You Very Much....

**ebaines** · July 23rd 2012, 12:53 PM

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.

**kjchauhan** · July 23rd 2012, 04:31 PM

Yes, I typed correctly as given in the book, may be it was printed wrong in book, thanks ..

**Amer** · July 23rd 2012, 06:50 PM

Originally Posted by kjchauhan

Please Help me to prove this,

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

Thank You Very Much....

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

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

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

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

You can make the nominator and denominator with common denominator then cancel it

**kjchauhan** · July 23rd 2012, 11:38 PM

Thank you very much for giving such an important hint..

**ebaines** · July 24th 2012, 05:08 AM

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.

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