Various Forms Of The Equation Of A Line

We are asked to simplify the expression . To solve this, we can use trigonometric identities and relationships between the angles. Start by noting that: This result comes from the fact that the product of cosines of angles which sum up to specific angles (like the sum of ) can be simplified using known identities or patterns that relate the angles to sine and cosine values.
Thus, the correct answer is option (D), .

We are asked to evaluate the sum:

Step 1: Use the periodicity of powers of . The powers of repeat every four terms:

Thus, the powers of repeat every four terms.

Step 2: Notice that will follow a periodic pattern as well. Let's compute the first few terms to identify the pattern:

- For , ,
- For , ,
- For , ,
- For , .

Step 3: The sum follows the pattern , which repeats every four terms.

Step 4: Since the sum repeats every 4 terms and we are summing from to , we have 6 full cycles of the 4-term pattern. The sum of one full cycle is:

Thus, each cycle contributes 0 to the sum.

Step 5: Since there are 6 full cycles, the total sum is:

Thus, the value of the sum is .

Therefore, the correct answer is option (D).

The equation of the line through two points and can be written as: Substituting the coordinates of and : This simplifies to: Thus, the equation of the line is:

The equation of the line through two points and can be written as: Substituting the coordinates of and : This simplifies to: Thus, the equation of the line is:

Step 1: Understanding the Concept:
A line's equation can be found using two points. Once found, its coefficients can be compared to the given form .
Step 2: Key Formula or Approach:
Slope
Point-slope form:
Step 3: Detailed Explanation:
Points: and .
.
Equation: .
Comparing with :
.
Value of .
Step 4: Final Answer:
The value of is 5.