This is a controversial subject and there will be many opinions agreeing or disagreeing with the essence and reason for happiness. Happiness is so subjective that defining it is almost impossible. However, for this particular argument - to find three proofs for your position - as to why it is necessary to believe in God in order to be happy - examine human nature.
People often talk about "searching" for something missing in their lives. It is their quest for happiness. When people are not happy, they need to find an outside source of any unhappiness in order to experience personal development. Blaming themselves is counter-productive and blaming the world at large is easier than self-reflection. To prevent aggressive, vengeful and hateful behavior, they need to focus on something good. For the purposes of this argument, that would be God.
1. We need God in our lives in order to be happy so that we can always relate to an outside source of goodness, even when things go wrong and this will always uplift our spirits, making us "happy," even if the happiness is transient Having God encourages us to keep striving towards an inner happiness. There is much evil in the world and people need to find a reason to explain this circumstance. People can blame bad actions on a lack of religion rather than human greed.
2. We need God in our lives to ease the pain of disappointment and failure. We need to be able to rationalize evil and take comfort in the knowledge that, failing punishment on earth, evil-doers will be justly punished by God. This enables us to go on with our lives after a traumatic event and allows us to continue striving for happiness. Without that belief in God, living may become futile. Having no concept of God may prevent people from seeing any kind of future for themselves.
As the main principle of all religions is faith, a belief in God makes us happy as we can trust Him if not others. Many things in life do not make sense so we need faith in order to accept and make the most out of any situation - thereby gaining happiness.
3. God gives us direction and he allows us to make mistakes and still recover from them and go on to be better people. This in turn makes us happy! What happiness the Prodigal Son and his father experienced on the son's return. We receive second chances from God that allow us to continue on and seek to be happy.
Happiness is only assured if a person is content and, in this world, it is difficult to find contentment when there is so much grief and war and inflicted trauma. Thus we need to have a higher purpose, to believe in something not of this earth so that we can make our own valuable contribution. It is this contribution that leads to happiness and for many, the avenue of contribution comes through the knowledge of God.