If the resultant of two vectors each of magnitude F is also of magnitude F, the angle between them will be?

There are two vectors. The magnitude of each vector is F.
Suppose the first vector lies horizontally and point right, and the second vector makes an angle θ with the first one.

The horizontal component of the resultant vector = F + F cosθ
The vertical component of the resultant vector = 0 + F sinθ = F sinθ

The magnitude of the resultant vector is F :
(F + F cosθ)² + (F sinθ)² = F
(1 + cosθ)² + sin²θ = 1
1 + 2 cosθ + cos²θ + sin²θ = 1
1 + 2 cosθ + (cos²θ + sin²θ) = 1

But sin²θ + cos²θ = 1, then :
1 + 2 cosθ + 1 = 1
1 + 2 cosθ = 0
cosθ = -1/2
θ = 180° - 45°
θ = 135°

The angle between the two vectors = 135°

The resultant of two vectors A and B is
R^2 =A^2 + B^2 + 2 AB cos theta.
Here R=F= A =B,
F^2 = 2 F^2 (1+ cos theta),
1+ cos theta = 1/2, cos theta = -1/2, theta = pi+60= 240 degrees