1. Condition for a body to just begin sliding on an incline

When a block is on an inclined plane of angle $\theta$, the forces along the plane are:

• Component of weight down the plane: $mg\sin\theta$
• Maximum static friction up the plane: $f_s^{\max}=\mu_s N$
• Normal reaction: $N=mg\cos\theta$

At the point where the block just begins to slide, limiting equilibrium gives:

$mg\sin\theta = \mu_s mg\cos\theta$

So,

$\mu_s = \tan\theta$

2. Substitute the given angle

Here,

$\theta=45^\circ$

Hence,

$\mu_s = \tan 45^\circ = 1$

3. Match with the options

• A: $1$ ✅
• B: $1.7$ ❌
• C: $0.5$ ❌
• D: $\dfrac{1}{\sqrt{3}}$ ❌

Therefore, the correct answer is:

$\boxed{1}$

So, Option A is correct.