What is the average of the sixty terms given below?
`cos^(2)x, cos^(2)2x, cos^(2)3x, …….. , cos^(2) 30 x, sin^(2)x, sin^(2) 2x, sin^(2) 3x, ………, sin^(2) 30x`

Options

A. `cos^(2)x`
B. `0.5`
C. `1`
D. `cos^(2)x sin^(2)x`

Answer

To find the average of the given 60 terms, we will follow these steps:

1. **Identify the Terms**: The terms are \( \cos^2 x, \cos^2 2x, \cos^2 3x, \ldots, \cos^2 30x \) and \( \sin^2 x, \sin^2 2x, \sin^2 3x, \ldots, \sin^2 30x \).

2. **Count the Number of Terms**: There are 30 terms of \( \cos^2 \) and 30 terms of \( \sin^2 \), making a total of 60 terms.

3. **Calculate the Sum of the Terms**: We can group the terms as follows:
\[ \text{Sum} = (\cos^2 x + \sin^2 x) + (\cos^2 2x + \sin^2 2x) + \ldots + (\cos^2 30x + \sin^2 30x) \]

4. **Use the Pythagorean Identity**: We know that:
\[ \cos^2 \theta + \sin^2 \theta = 1 \]
Therefore, each pair \( (\cos^2 kx + \sin^2 kx) \) for \( k = 1, 2, \ldots, 30 \) equals 1.

5. **Calculate the Total Sum**: Since there are 30 pairs:
\[ \text{Sum} = 1 + 1 + 1 + \ldots + 1 \quad (\text{30 times}) = 30 \]

6. **Calculate the Average**: The average is given by the formula:
\[ \text{Average} = \frac{\text{Sum of all terms}}{\text{Number of terms}} = \frac{30}{60} = \frac{1}{2} = 0.5 \]

Thus, the average of the 60 terms is \( 0.5 \).