Proof: tan^-1(x) + tan^-1(y) = tan^-1 ((x+y)/(1-x*y))

[johnq2k7]

Prove tan^-1 (x) + tan^-1 (y)= tan^-1 ((x+y)/(1-x*y))

since, arctan(x)= tan^-1(x)=y

and tan y= x

therefore tan^-1(x) + tan^-1(y) = y+ tan^-1(y)
therefore tan^-1(x) + tan^-1(y) = y+ (tan (y))^-1
therefore tan^-1(x) + tan^-1(y)= y+ x^-1
therefore tan^-1 (x)+ tan^-1(y)= (y/x)

I think my approach and work here was wrong...

I think I need to use the tan(x+y) identity

but how do I go about using this identity?

I know tan (x+y) = (tan (x) + tan (y))/ (1- tan (x)+*tan(y))

however, how do I apply this identity considering

tan^-1 (x) + tan^-1 (y) is NOT EQUAL to tan^-1 (x+y)

please help me with this problem

 

[pka]

Re: Proof question!... work shown... need help!

Let \(\displaystyle a = \arctan (x)\;\& \;b = \arctan (y)\; \Rightarrow \;x = \tan (a)\;\& \;y = \tan (b)\).

\(\displaystyle \begin{gathered} \frac{{x + y}}{{1 - xy}} = \frac{{\tan (a) + \tan (b)}}{{1 - \tan (a)\tan (b)}} = \tan (a + b) \hfill \\ \hfill \\ \end{gathered}\)

Can you finish?

 

[Deleted member 4993]

Re: Proof question!... work shown... need help!

Prove tan^-1 (x) + tan^-1 (y)= tan^-1 ((x+y)/(1-x*y))

since, arctan(x)= tan^-1(x) = y <<<< Why is that?

assume

tan[sup:4jy1v0xo]-1[/sup:4jy1v0xo](x) = a and

tan[sup:4jy1v0xo]-1[/sup:4jy1v0xo](y) = b

We have

tan[tan[sup:4jy1v0xo]-1[/sup:4jy1v0xo] ((x+y)/(1-x*y))] = (x+y)/(1-x*y)

Then

(x+y)/(1-x*y)

= [tan(a) + tan(b)]/[1 - tan(a) * tan(b)]

Now continue....

 

[johnq2k7]

Re: Proof question!... work shown... need help!

Let \(\displaystyle a = \arctan (x)\;\& \;b = \arctan (y)\; \Rightarrow \;x = \tan (a)\;\& \;y = \tan (b)\).

\(\displaystyle \begin{gathered} \frac{{x + y}}{{1 - xy}} = \frac{{\tan (a) + \tan (b)}}{{1 - \tan (a)\tan (b)}} = \tan (a + b) \hfill \\ \hfill \\ \end{gathered}\)

Can you finish?

I think I know how to complete the rest but I'm not sure my mathematical approach is completely correct, here is my work:

since (x+y)/(1-x*y)= tan (a+b)

and since a= tan^-1(x) and b= tan^-1(y)

therefore tan^-1 ((x+y)/(1-x*y))= tan^-1(tan(tan^-1(x)+ tan^-1(y))

since tan^-1(x)*(tan)= x

therefore tan^-1((x+y)/(1-x*y))= tan^-1(x)+tan^-1(y)

is this correct?