use below code to add product to cart

       try
    {
    
    	$cart = Mage::getSingleton('checkout/cart');
        $cart->init();
    
        $qty=1;
    
        $productId='10'; //product id
    
        $product = Mage::getModel('catalog/product')->load($productId);
        $formKey = Mage::getSingleton('core/session')->getFormKey();
    
        if($product->getSku()!='')
        {
            $paramater = array(
                'product' => $productId,
                'qty' => $qty,
                'form_key' => $formKey
            );
    
            $cart->addProduct($product, $paramater);
            $cart->save();
        }
        else
        {
            Mage::log('product not vailable');  
        }
        $cart->save();
    	Mage::getSingleton('checkout/session')->setCartWasUpdated(true);
    }catch (Exception $e)
    {
         Mage::log($e->getMessage());
    }