Question: I am having a confusion whether an opportunity can exists without products. If such an opportunity exists then what would be the use of it in the context of business perspective? If i recall correctly that an opportunity amount is the sum of the cost price of the opportunity line items, is that correct? Any suggestions please.

Answer: 
Opportunities without products are a normal use case.

For example, a business that doesn't really care to forecast by product as it develops 'opportunities' - simply knowing the customer name + amount + stage + close date is sufficient.

The Opportunity.amount must be entered by the user rather than calculated as a RSF on OpportunityLineItem.
Yes, an opportunity can exist without products but the Amount would need to either be manually populated or populated via a workflow rule/trigger(only auto updates when products do exist). We do not use the standard products and use a workflow rule to update the Amount field based on a rollup of a custom object.