Expert: Paul Klarreich - 9/24/2009

Question
Show that: y= 4x^2 (1-x^2) if sin^-1 y= 2cos^-1 x

Answer
Questioner: Edwin
Country: Norway
Category: Advanced Math
Private: No
Subject: Math
Question: Show that: y= 4x^2 (1-x^2) if sin^-1 y= 2cos^-1 x

>>> Misprint; That should say:

Show that: y^2 = 4x^2 (1-x^2)
....................................
Hi, Edwin,

I shall rewrite the condition:

sin^-1 y= 2cos^-1 x   

as

arcsin y = 2 arccos x

Now arc<anything>(x or y)  is an angle.
so write  arccos x = t, and that means

cos t = x,  also
cos^2 t = x^2, also
sin^2 t = 1 - cos^2 t = 1 - x^2

Then  arcsin y = 2t and that means  y = sin(2t)

y = sin(2t) = 2 sin t cos t
y^2 = (2 sin t cos t)^2

y^2 = 4 sin^2 t cos^2 t

y^2 = 4 (1 - x^2) x^2

which is what you had to prove?