Why use the income approach to approximate GDP if it relies on the value produced by the expenditures approach? To preface, I'm taking a macroeconomics class and we're learning about measuring GDP. The textbook says that there are two main ways to measure GDP: the expenditures method and the income method.

The expenditures method can be represented like so:

$GDP = C + I_g + G + X_n$

where $C$ is personal consumption expenditures, $I_g$ is gross private domestic investment, $G$ is government purchases, and $X_n$ is net exports.

The income method is more complicated, and looks like this:

$GDP = I_n - I_f - C_c + s$

Where $I_n$ is the national income, $I_f$ is the net foreign income, $C_c$ is the consumption of fixed capital, and $s$ is the statistical discrepancy.

$s$ is calculated by subtracting the expenditures method from the income method, in order to make them the same. 

What is the point of having the income method if it relies on the expenditures method? I get that it would be useful for statistics, but why use both?
The definition of the expenditure measure of GDP is:
$$GDP_{EXP} \equiv C + I_g + G + X_n$$

The definition of the expenditure income measure of GDP is:
$$GDP_{income} \equiv I_n - I_f - C_c$$

As I understand it, $s$ is not part of it. $s$ is defined as the difference between the two measures:
$$s \equiv GDP_{EXP} - GDP_{income}$$

Apparently it is also possible to define it with the RIPSAW approach:
$$GDP_{income} \equiv R + I + P + S + A + W$$
R = rents, I = interests, P = profits, SA = statistical adjustments (corporate income taxes, dividends, undistributed corporate profits, and W = wages.

And yes, it is useful to calculate GDP with multiple methods. Among other things, it reduces noise in GDP estimates:


  Two often-divergent U.S. GDP estimates are available, a widely-used
  expenditure-side version $GDP_E$ , and a much less widely-used
  income-side version $GDP_I$ . We propose and explore a “forecast
  combination” approach to combining them. We then put the theory to
  work, producing a superior combined estimate of GDP growth for the
  U.S., $GDP_C$ . We compare $GDP_C$ to $GDP_E$ and $GDP_I$ , with particular
  attention to behavior over the business cycle. We discuss several
  variations and extensions


Improving U.S. GDP Measurement: A Forecast Combination Perspective Aruoba, Diebold, Nalewaik, Schorfheide, and Song (2012, ungated here)