Check $X$:

$X:a\lor(b\land(a\lor c))=(a\lor b)\land(a\lor c)$

This follows the distributive law:

$a\lor(b\land d)=(a\lor b)\land(a\lor d)$

Here,

$d=(a\lor c)$

So,

$a\lor(b\land(a\lor c))=(a\lor b)\land(a\lor(a\lor c))$

$=(a\lor b)\land(a\lor c)$

Therefore, $X$ is true.

Now check $Y$:

$Y:a\land(b\lor(a\land c))=(a\land b)\lor(a\land c)$

Using distributive law:

$a\land(b\lor d)=(a\land b)\lor(a\land d)$

Here,

$d=(a\land c)$

So,

$a\land(b\lor(a\land c))=(a\land b)\lor(a\land a\land c)$

$=(a\land b)\lor(a\land c)$

Therefore, $Y$ is true.

Hence, both $X$ and $Y$ are true.

Hence, the correct answer is $\boxed{(c)}$