Simplyfy csc2arcsinxSolutioncsc 2 arc sin x Let 2 arc sin xt

Simplyfy csc(2arcsin(x))

Solution

csc (2 arc sin x)

Let 2 arc sin x=t

arc sin x=t/2

x=sin(t/2)

sin(t/2)= (sqrt(1-cos t)/2)

x²=1-cos t/2

1-cos t=2x²

cos t= 1-2x²

sin t= sqrt(1-(1-2x²)²)/1

sin t= 1/(-4x²(x²+1))= 2xsqrt(- x²+1)

csc t=1/sin t= 1/2xsqrt(-x²+1)