This is one of the conventional themes in macroeconomics and most of students in introductory economics class may study which policy is more effective on the stagnant economy, tax cut or government spending.
As introductory macroeconomics(say, the Keynesian Cross) says, government spending is more influential, more likely to increase the national income(GDP).
The reason is quite simple: government does a debt-finance.
Imagine two people earn income of $100 a day, buy food from each other at a cost of $10 and the income tax imposed is $10. Thus each of them has $90 in hand and spends $10 on food a day.
In this economy, the total income (the sum of two people's incomes) is $180.
Due to economic stagnation, one of them becomes unemployed and earns $0 a day. If the government cuts income tax (-$10) to stimulate the economy, the employed will have more(she has now $100 in hand a day) but the unemployed will not. So the unemployed has no food to eat this day. (In this case the unemployed can earn money to sell food to the employed, because both eat nothing.)
In this tax-cut economy, the total income is $100.
Alternatively, if the government provides the unemployed with a job that makes him earn $10 a day, that is, increases the government spending, then the unemployed will get the benefit(he has now $10 a day in hand) but the employed will not (her income remains $90 in hand) .
In such a spending economy, the total income is $100. The same! This is not more effective than tax cut.
The reason is in government-spending economy, she is still imposed income tax of $10. The point is, in a tax-cut economy, income distribution is done through the exchange from each other, whereas in a spending economy through the government that just takes $10 from the employed and gives $10 to the unemployed.
If there were no income tax on her(government made a policy of debt-finance) in a spending economy, the total income would be $110 and thus more effective than tax cut policy (of course, later a tax increase will come). Keynesian policy is always a debt-finance policy!