Let there are two vectors  A and B.

To find the resultant (A+B) , we will use the  triangle law of vector addition.

To get resultant as (A-B), we will reverse the vector B. As A+(-B)= A-B.

See figure (1)

So,let the angle between vector A and resultant (A+B) is θ₁

angle between vector A and (A-B) is θ₂

Now, we have two new vectors, vector (A+B) and vector (A-B)       as shown in figure (2)

So, using the parallelogram law of vector addition, when we arrange these two vectors, we will get their resultant at the diagonal of parallelogram, which is= (A+B)+(A-B) = 2A

This resultant vector 2A, is double od vector V.

So the angle between the resultant of (A+B) and (A-B) and vector A is 0.

But the magnitude is double.