Simplify

Hey /sci/

I was wondering if any of you could help me simplify:
sin⁡(2 csc^(-1)⁡x)

i see boxes.  ⁡

sin(2 csc^-1(x))

How bout now?
It would be most beneficial if you showed the steps too so I could reproduce the work for different problems

anybody know how?
sin(2arccsc x)

sin (2c^2ra)

I dunno. Draw a triangle maybe.

t = arcsec(x)
Sec(t) = x
Cos(t) = 1/x

Sin(t)^2 + Cos(t)^2 = 1
Sin(t) = sqrt[1-cos(t)^2]
Sin(t) = sqrt(1 - 1/x^2)
Sin(t) = sqrt(x^2 - 1)/x

sin(2t) = 2*Cos(t)*Sin(t)
sin(2t) = 2*sqrt(x^2 - 1)/x^2

>>1
= sin (2 csc^(-1) x)
= sin (2 (-i ln(i/x + sqrt(1 - 1/x^2))))
= sin (-2i ln(i/x + sqrt(1-1/x^2)))
= (e^(i(-2i ln(i/x + sqrt(1-1/x^2)))) - e^(-i(-2i ln(i/x + sqrt(1-1/x^2)))))/(2i)
= ((i/x + sqrt(1-1/x^2)) e^2 - (i/x + sqrt(1-1/x^2)) e^(-2))/(2i)
= (i/x + sqrt(1-1/x^2)) (e^2-e^(-2))/(2i)
= (i/x + sqrt(1-1/x^2))/(2i)
= 1/(2x) + sqrt(1-1/x^2)/(2 sqrt(-1))
= 1/(2x) + sqrt((1-1/x^2)/(-1))/2 <----can I do this?
= 1/2 (x + sqrt(1/x^2 - 1))

Is this correct?

>>8
Probably not.
Plug x=2 into the original equation.  You get sqrt(3)/2 out.
Plug x=2 into your equation.  You get a complex number out.
Plug x=2 into >>7's equation.  You get sqrt(3)/2 out.

>>8
And no, you can't do that.  You have to bring out a factor of -1.
-sqrt(1/-1) = -sqrt(-1) = -i = 1/i = 1/sqrt(-1) = sqrt(1)/sqrt(-1)

>>8
= (e^(i(-2i ln(i/x + sqrt(1-1/x^2)))) - e^(-i(-2i ln(i/x + sqrt(1-1/x^2)))))/(2i)
= ((i/x + sqrt(1-1/x^2)) e^2 - (i/x + sqrt(1-1/x^2)) e^(-2))/(2i)

phail
it should be (i/x + sqrt(1-1/x^2))² - (i/x + sqrt(1-1/x^2))^-2

Also, e² - e^-2 > 2² - e^-2 = 4 - e^-2 > 4 - 1 = 3