Why do we need a voltage divider when we get the same voltage at the output as the input?

The 10k\$\Omega\$ resistor is there to pull down the gate when the input is floating (thus avoiding an undefined/uncontrolled gate voltage). On the other hand, the 100\$\Omega\$ resistor is there to limit gate charging/discharging current (due to the presence of gate capacitance) and to prevent oscillations.

But, as you have already detected, both resistors form a voltage divider. Thus, we need to avoid dropping too much voltage in the 100\$\Omega\$ resistor, otherwise we might not be able to achieve the gate voltage required to turn ON the MOSFET.

The solution is to let the pull down resistor be much, much higher than the gate resistor. 20x higher is often used; in your case it's 100x higher, which is fine too. The pull down resistor value is not critical and is usually something between 4.7k\$\Omega\$ and 100k\$\Omega\$.

Here that 10K Ohms acts as a pull down resistor only ie., to make the gate low normally. in otherwords , we are defining the FET to be in OFF condition normally.