(sqrt[1 + x^2] + sqrt[1 - x^2])/(sqrt[1 + x^2] - sqrt[1 - x^2]) = (1 + sqrt[1 - x^4])/x^2√1+x2+√1−x2√1+x2−√1−x2=1+√1−x4x2 or
phi = tan^-1( (1 + sqrt[1 - x^4])/x^2)ϕ=tan−1(1+√1−x4x2) or
(1 + sqrt[1 - x^4])/x^2 = tan phi1+√1−x4x2=tanϕ or
(x^2 tan phi-1)^2= 1-x^4(x2tanϕ−1)2=1−x4 or
x^4 tan^2phi-2 x^2 tan phi+1 = 1-x^4x4tan2ϕ−2x2tanϕ+1=1−x4 or
x^2 = sin(2phi)x2=sin(2ϕ) or
phi = 1/2 sin^-1(x^2)ϕ=12sin−1(x2)
The last step is left to the reader as an exercise. Be careful because the squaring operation can introduce strange solutions.
NOTE
2phi = sin^-1(x^2) rArr 2phi = pi/2-cos^-1(x^2) rArr phi = pi/4-1/2cos^-1(x^2)2ϕ=sin−1(x2)⇒2ϕ=π2−cos−1(x2)⇒ϕ=π4−12cos−1(x2)
