Cos Series Value: Understanding the Problem

The problem asks us to find the sum of a series involving cosine squared terms: \( \cos^2 1^\circ + \cos^2 5^\circ + \cos^2 9^\circ + \dots + \cos^2 81^\circ + \cos^2 85^\circ + \cos^2 89^\circ \). To solve this cos series, we need to analyze the pattern of the angles and apply relevant trigonometric identities.

Angles Pattern: Identifying Arithmetic Progression

Let's examine the angles in the given cos series: \(1^\circ, 5^\circ, 9^\circ, \dots, 81^\circ, 85^\circ, 89^\circ\).

• The first term is \(a_1 = 1^\circ\).
• The second term is \(a_2 = 5^\circ\).
• The third term is \(a_3 = 9^\circ\).

We can observe that the difference between consecutive terms is constant:

• \(5^\circ - 1^\circ = 4^\circ\)
• \(9^\circ - 5^\circ = 4^\circ\)

This confirms that the angles form an arithmetic progression with a common difference \(d = 4^\circ\). The last term in this arithmetic progression is \(a_n = 89^\circ\).

Number of Terms: Calculating 'n'

To find the total number of terms (\(n\)) in the cos series, we use the formula for the n-th term of an arithmetic progression:

\(a_n = a_1 + (n-1)d\)

Substituting the values:

\(89^\circ = 1^\circ + (n-1)4^\circ\)

\(88^\circ = (n-1)4^\circ\)

\(22 = n-1\)

\(n = 23\)

Therefore, there are 23 terms in the given cos series.

Trigonometric Identities: Key to Simplification

We will use the following fundamental trigonometric identities to simplify the sum of the cos series:

• Complementary Angle Identity: \( \cos (90^\circ - \theta) = \sin \theta \)
• Pythagorean Identity: \( \cos^2 \theta + \sin^2 \theta = 1 \)

Combining these, we can write:

\( \cos^2 \theta + \cos^2 (90^\circ - \theta) = \cos^2 \theta + \sin^2 \theta = 1 \)

This identity is crucial for pairing the terms in our cos series.

Pairing Terms: Simplifying the Sum

Since there are 23 terms, which is an odd number, there will be one middle term that does not have a pair. The middle term is the \(\left(\frac{n+1}{2}\right)\)-th term, which is the \(\left(\frac{23+1}{2}\right)\)-th = 12th term.

Let's find the 12th term in the arithmetic progression of angles:

\(a_{12} = a_1 + (12-1)d = 1^\circ + 11 \times 4^\circ = 1^\circ + 44^\circ = 45^\circ\)

So, the middle term is \( \cos^2 45^\circ \).

Now, let's pair the terms from the beginning and the end of the cos series using the identity \( \cos^2 \theta + \cos^2 (90^\circ - \theta) = 1 \):

Since there are 23 terms and one middle term (\( \cos^2 45^\circ \)), the remaining \(23-1=22\) terms form \(\frac{22}{2} = 11\) pairs. Each of these 11 pairs sums to 1.

So, the sum of these 11 pairs is \(11 \times 1 = 11\).

Middle Term Calculation: \(\cos^2 45^\circ\)

The value of \( \cos 45^\circ \) is \( \frac{1}{\sqrt{2}} \).

Therefore, the middle term \( \cos^2 45^\circ \) is:

\( \cos^2 45^\circ = \left(\frac{1}{\sqrt{2}}\right)^2 = \frac{1}{2} \)

Final Series Sum: Combining Results

The total sum of the cos series is the sum of the paired terms plus the middle term:

Total Sum = (Sum of 11 pairs) + (Value of middle term)

Total Sum = \(11 + \frac{1}{2}\)

Total Sum = \(11\frac{1}{2}\)

Thus, the value of the given cos series is \(11\frac{1}{2}\).