Question

Write the given expression in terms of x and y only sin(tan^-1(x)+tan^-1(y)

Answer 1

Answer:

Step-by-step explanation:

Let a = tan⁻¹(x), so tan(a) = x,

$\sin(a)=\frac{x}{\sqrt{1+x^2}}, \cos(a)=\frac{1}{\sqrt{1+x^2}},$

Let b = tan⁻¹(y), so tan(b) = y,

$\sin(b)=\frac{y}{\sqrt{1+y^2}}, \cos(b)=\frac{1}{\sqrt{1+y^2}},$

sin( tan⁻¹(x) +  tan⁻¹(y)) = sin(a + b)

= sin(a) cos(b) + cos(a)sin(b)

= $\frac{x}{\sqrt{1+x^2}}\times\frac{1}{\sqrt{1+y^2}}+ \frac{y}{\sqrt{1+y^2}}\times\frac{1}{\sqrt{1+x^2}}$

$=\frac{x+y}{\sqrt{(1+x^2)(1+y^2)}}$