Question

Solve the equation tan-1(1+x/1-x)=π/4+tan-1x

Answer 1

Given:  The expression tan^-1 (1+x/1-x) = π/4 + tan^-1 x

To find: Solve the equation .

Solution:

• Now we have given the expression as:

                  tan^-1 (1+x/1-x) = π/4 + tan^-1 x

• Taking tan^-1 terms on one side, we get:

                  tan^-1 (1+x/1-x) - tan^-1 x = π/4

• Now we know the formula :

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

• Using this formula, we get:

                  tan^-1 ((1+x/1-x) - x / 1 + (1+x/1-x)x) ) = π/4

                  (1+x/1-x) - x / 1 + (1+x/1-x)x)  = tan π/4

                  (1 + x - x + x^2 ) / 1-x / (1 - x + x + x^2) / 1-x = 1

• Cancelling 1 - x and solving further, we get:

                  1 + x^2 / 1 + x^2 = 1

                  1 = 1

• This is independent of x.

• Hence given equation will be satisfied for all x∈(0,1).

Answer:

               So the given equation will be satisfied for all x∈(0,1).