The value of `sum_(r=1)^(89)log_(10)(tan((pir)/180))` is equal to

Options

A. 10
B. 1
C. 0
D. none of these

Explanation

To solve the problem, we need to evaluate the summation:

\[ y = \sum_{r=1}^{89} \log_{10} \left( \tan\left(\frac{\pi r}{180}\right) \right) \]

### Step 1: Rewrite the Summation
We can use the property of logarithms that states \(\log(a) + \log(b) = \log(ab)\). Therefore, we can rewrite the summation as:

\[ y = \log_{10} \left( \prod_{r=1}^{89} \tan\left(\frac{\pi r}{180}\right) \right) \]

**Hint:** Remember that the logarithm of a product is the sum of the logarithms.

### Step 2: Analyze the Tangent Function
Notice that \(\tan\left(\frac{\pi r}{180}\right)\) has a symmetry property. Specifically, we have:

\[ \tan\left(\frac{\pi (90 - r)}{180}\right) = \cot\left(\frac{\pi r}{180}\right) \]

This means that for every \(r\) from 1 to 89, there is a corresponding \(90 - r\) such that:

\[ \tan\left(\frac{\pi r}{180}\right) \cdot \tan\left(\frac{\pi (90 - r)}{180}\right) = 1 \]

**Hint:** Look for pairs in the summation that can simplify the product.

### Step 3: Group the Terms
We can group the terms in pairs:

\[ \tan\left(\frac{\pi r}{180}\right) \cdot \tan\left(\frac{\pi (90 - r)}{180}\right) = 1 \]

This means that for \(r = 1\) to \(44\), each pair contributes a product of 1. The middle term when \(r = 45\) is:

\[ \tan\left(\frac{\pi \cdot 45}{180}\right) = \tan\left(\frac{\pi}{4}\right) = 1 \]

Thus, the entire product simplifies to:

\[ \prod_{r=1}^{89} \tan\left(\frac{\pi r}{180}\right) = 1 \]

**Hint:** Consider how symmetry in the tangent function can help simplify the product.

### Step 4: Substitute Back into the Logarithm
Now substituting back into the logarithm, we have:

\[ y = \log_{10}(1) = 0 \]

**Hint:** Remember that the logarithm of 1 in any base is always 0.

### Final Answer
Thus, the value of the summation is:

\[ \boxed{0} \]