Question

If x=ln((tan(45+(y/2))) show that sinh x=tan y

Answer 1

tan(45 + y/2) = (tan45 + tan(y/2))/(1 - tan45 tan(y/2))

= (1+tan(y/2))/(1-tan(y/2))
so,
e^x = (1+tan(y/2)) / (1 - tan(y/2))
e^-x = (1 - tan(y/2)) / (1+tan(y/2)))

Now, recall that sinhx = (e^x - e^-x)/2
So, letting z = tan(y/2) for ease of typing,
sinhx = (1+z)/(1-z) - (1-z)/(1+z)
= ((1+z)^2 - (1-z)^2)/(1-z^2)
= 2z/(1-z^2)
But, since z = tan(y/2), that is just tan(y)