about multiple angles of trigonometry

my answer is different from the provided answer.

can someone please help me to check where i gone wrong??

the question is:

given that sin X= - 5/13, cos Y = - 4/5 and that X and Y are in the same quadrant, find the value of sin(X/2).

my solution :

cos X = 2 cos^2 (X/2) -1

cos (X/2) = square root [(1/13)/2]

cos (X/2)= 1/(square root 26)

cos (X/2)= (square root 26)/26

then, sin X = 2(sin X/2)(cos X/2)

-5/13=2(sin X/2)[(square root 26)/26]

-5/13 x [26/(square root 26)] = 2(sin X/2)

[-10/(square root 26)] x 1/2 = sin X/2

sin (X/2) = -5/(square root 26)

rationalise

sin (X/2) = [(-5square root 26)/26]

but the provided answer is (5square root 26)/26 .....

which means mine is negative but the provided answer is positive...

can anyone tell me where i gone wrong???