Display Shipping Cost in Cart Without Page Reload

Code

```php
// Enqueue custom script for AJAX shipping cost update
function wp_dudecom_enqueue_ajax_shipping_script() {
    if (is_cart()) {
        wp_enqueue_script('wp-dudecom-ajax-shipping', get_template_directory_uri() . '/js/wp-dudecom-ajax-shipping.js', array('jquery'), null, true);
        wp_localize_script('wp-dudecom-ajax-shipping', 'wp_dudecom_ajax_shipping_params', array(
            'ajax_url' => admin_url('admin-ajax.php'),
            'nonce'    => wp_create_nonce('wp_dudecom_ajax_shipping_nonce')
        ));
    }
}
add_action('wp_enqueue_scripts', 'wp_dudecom_enqueue_ajax_shipping_script');

// AJAX handler to calculate shipping cost
function wp_dudecom_calculate_shipping_cost() {
    check_ajax_referer('wp_dudecom_ajax_shipping_nonce', 'security');

    if (WC()->cart->needs_shipping()) {
        WC()->cart->calculate_shipping();
        WC()->cart->calculate_totals();

        $packages = WC()->shipping->get_packages();
        $shipping_cost = '';

        foreach ($packages as $package) {
            if (!empty($package['rates'])) {
                foreach ($package['rates'] as $rate) {
                    $shipping_cost = wc_price($rate->cost);
                    break;
                }
            }
        }

        wp_send_json_success(array('shipping_cost' => $shipping_cost));
    } else {
        wp_send_json_error('No shipping needed');
    }
}
add_action('wp_ajax_wp_dudecom_calculate_shipping_cost', 'wp_dudecom_calculate_shipping_cost');
add_action('wp_ajax_nopriv_wp_dudecom_calculate_shipping_cost', 'wp_dudecom_calculate_shipping_cost');

// Add custom script to handle AJAX request
add_action('wp_footer', 'wp_dudecom_add_ajax_shipping_script');
function wp_dudecom_add_ajax_shipping_script() {
    if (is_cart()) {
        ?>
        <script type="text/javascript">
            jQuery(document).ready(function($) {
                function updateShippingCost() {
                    $.ajax({
                        url: wp_dudecom_ajax_shipping_params.ajax_url,
                        type: 'POST',
                        dataType: 'json',
                        data: {
                            action: 'wp_dudecom_calculate_shipping_cost',
                            security: wp_dudecom_ajax_shipping_params.nonce
                        },
                        success: function(response) {
                            if (response.success) {
                                $('.shipping-cost').text(response.data.shipping_cost);
                            } else {
                                console.log(response.data);
                            }
                        }
                    });
                }

                // Trigger shipping cost update on cart page load
                updateShippingCost();

                // Optionally, trigger on other events like quantity change
                $(document.body).on('updated_cart_totals', function() {
                    updateShippingCost();
                });
            });
        </script>
        <?php
    }
}
```

Note: This code assumes you have a `.shipping-cost` element in your cart template where the shipping cost will be displayed. Adjust the selector as needed to match your theme's HTML structure.