Question

Prove the identity. [Hint: Let u = tan−1(x) and v = tan−1(y), so that x = tan(u) and y = tan(v). Use an Addition Formula to find tan(u + v).] (Enter your answers in terms of u and v.) tan−1 x + y 1 − xy = tan−1(x) + tan−1(y), −1 < x < 1, −1 < y < 1

Answer 1

Answer with Step-by-step explanation:

LHS

`tan^(-1)x+tan^(-1)y`

`u=tan^(-1)x, v=tan^(-1)y`

`tan(u+v)=(tanu+tanv)/(1-tanutanv)`

By using the formula

`tan(x+y)=(tanx+tany)/(1-tanxtany)`

Substitute the values

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

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

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

Hence, proved.