Let $\vec a$ and $\vec b$ be two unit vectors.

Statement-1: $|\vec a+\vec b|+|\vec a −\vec b|=|\vec a| + |\vec b|$

Statement-2: The greatest value $|\vec a+\vec b|+|\vec a −\vec b|$ is $2\sqrt{2}$

Statement-1 is False, Statement-2 is True.

Clearly, statement-1 is not true.

Let θ be the angle between unit vectors $\vec a$ and $\vec b$. Then,

$\vec a.\vec b=\cos θ$

Now,

$|\vec a+\vec b|^2$

$=|\vec a|^2+|\vec b|^2+2\vec a.\vec b=2+2\cos θ=4\cos^2\frac{θ}{2}$

and,

$|\vec a-\vec b|^2=|\vec a|^2+|\vec b|^2-2\vec a.\vec b=2-2\cos θ=4\sin^2\frac{θ}{2}$

$⇒|\vec a+\vec b|=2\cos\frac{θ}{2},|\vec a-\vec b|=2\sin\frac{θ}{2}$

$⇒|\vec a+\vec b|+|\vec a-\vec b|=2(\cos\frac{θ}{2}+\sin\frac{θ}{2})≤2\sqrt{2}$

So, statement-2 is true.