number of real solutions of equation(1+ cos2x)1/2 = 21/2 sin inverse sinx,-Π <= x <=Π isans: 2

See

1+cos2x= 2cos²x;

using this and putting it in given expression I will get cosx = sin^-1sinx.   ,-Π <= x <= Π

Now  draw the curves of cosx  for  -Π <= x <= Π  and also draw the curve of sin^-1sinx for -Π <= x <= Π and find the no. of intersection points of the curve. They will intersect at two points only. So, Ans is 2.