1. The problem statement, all variables and given/known data Simplify the expression: cos(2sin^-1 (5x)) 2. Relevant equations Fundamental identities: 1 = sin^2 ϑ + cos^2 ϑ : I think you use this one? 3. The attempt at a solution Let y=2sin^-1(5x) sin y = 10x so, you plug in? 1 = 10x^2 + cos^2 y not really sure if im on the right path or what to do next
cos 2t = 1 - 2sin²t let t = sin^-1 5x so sin t = 5x cos 2t = 1 - 2(5x²) cos t = ( 1 - 2(5x²) ) / (2) is this correct?
cos(2t)=1-2(5x^{2}) is almost correct; if t=arcsin(5x), what is sin^{2}t ? But your last line is not correct. (Can you see why?) However, you don't need the last line; you have simplified the expression, and you're done!
Oh duh, ok from the start: cos(2 arcsin 5x) Let t = arcsin 5x so, sin t = 5x Since cos 2t = 1 - 2sin²t cos 2t = 1 - 2(5x)² cos 2t = 1 - 2(25x²)