Compute \(\tan \left( \tan^{-1} \frac {1}{3} - \tan^{-1} \frac{1}{5} + \tan^{-1} \frac{1}{7} \right)\)
+23246 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
