The value of `|(cos.(2pi)/(63),cos.(3pi)/(70),cos.(4pi)/(77)),(cos.(pi)/(72),cos.(pi)/(40),cos.(3pi)/(88)),(1,cos.(pi)/(90),cos.(2pi)/(99))|` is equal to

Options

A. 0
B. 1
C. `2cos.(pi)/(9)`
D. `cos.(pi)/(11)`

Answer

To solve the determinant

\[ D = \begin{vmatrix} \cos\left(\frac{2\pi}{63}\right) & \cos\left(\frac{3\pi}{70}\right) & \cos\left(\frac{4\pi}{77}\right) \\ \cos\left(\frac{\pi}{72}\right) & \cos\left(\frac{\pi}{40}\right) & \cos\left(\frac{3\pi}{88}\right) \\ 1 & \cos\left(\frac{\pi}{90}\right) & \cos\left(\frac{2\pi}{99}\right) \end{vmatrix} \]

we will analyze the structure of the determinant and the properties of cosine.

### Step 1: Rewrite the cosine terms
We can express the cosine terms in terms of angles that are easier to manipulate. For example, we can use the identity:

\[ \cos(x) = \cos\left(\frac{n\pi}{m}\right) \text{ for appropriate } n \text{ and } m \]

### Step 2: Identify relationships among the terms
Notice that the angles in the cosine functions are fractions of \(\pi\). We can look for patterns or relationships among these angles.

### Step 3: Check for linear dependence
If any two rows (or columns) of the determinant are linearly dependent, the determinant will equal zero. We can check if any of the rows can be expressed as a linear combination of the others.

### Step 4: Calculate the determinant
Using properties of determinants, we can simplify our calculations. For instance, if we find that two rows are equal or proportional, the determinant will be zero.

### Step 5: Conclude the value of the determinant
After performing the calculations and checking for linear dependence, we conclude that the value of the determinant is:

\[ D = 0 \]

### Final Answer
Thus, the value of the determinant is \(0\).

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