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Compute \(\tan \left( \tan^{-1} \frac {1}{3} - \tan^{-1} \frac{1}{5} + \tan^{-1} \frac{1}{7} \right)\)

 Jul 12, 2020
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Let  A  =  tan-1(1/3)   --->   tan(A) = 1/3

Let  B  =  tan-1(1/7)   --->   tan(B) = 1/7

Let  C  =  tan-1(1/5)   --->   tan(C) = 1/5

 

Problem:  tan( A - C + B )   --->   tan( A + B - C )   --->   tan( (A + B) - C )

 

Since:  tan(X - Y)  =  [ tan(X) - tan(Y) ] / [ 1 + tan(X)·tan(Y) ]

 

With  X = A + B  and  Y = C   --->    tan( (A + B) - C )

                                                 =     [ tan(A + B) - tan(C) ] / [ 1 + tan(A + B)·tan(C) ]

 

Since:  tan(X + Y)  =  [ tan(X) + tan(Y) ] / [ 1 - tan(X)·tan(Y) ]

 

With  X = A  and  Y = B   --->   [ tan(A + B) - tan(C) ] / [ 1 + tan(A + B)·tan(C) ]

 

=   [ ( tan(A) + tan(B) ) / ( 1 - tan(A)·tan(B) )  - tan(C) ]  /  [1 + ( tan(A) + tan(B) ) / ( 1 - tan(A)·tan(B) )·tan(C) ]

=  [ ( 7 + 3 ) / ( 21 - 1 ) -  (1/5) ]  /  [ 1 + ( 7 + 3 ) / ( 21 - 1 )·(1/5) ]

=  ( 3 / 10) / ( 11 / 21 )

=  3/11

 Jul 12, 2020

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