`cos^(2)5^(@) +cos^(2)10^(@) + cos^(2) 15^(@) +…..+cos^(2) 85^(@) + cos^(2) 90^(@) `is equal to

Options

A. 10
B. `19/2`
C. `9/2`
D. `17/2`

Explanation

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

\[ \cos^2(5^\circ) + \cos^2(10^\circ) + \cos^2(15^\circ) + \ldots + \cos^2(85^\circ) + \cos^2(90^\circ) \]

### Step 1: Identify the pairs
Notice that the angles can be paired such that each pair sums to \(90^\circ\):
- \(\cos^2(5^\circ) + \cos^2(85^\circ)\)
- \(\cos^2(10^\circ) + \cos^2(80^\circ)\)
- \(\cos^2(15^\circ) + \cos^2(75^\circ)\)
- \(\cos^2(20^\circ) + \cos^2(70^\circ)\)
- \(\cos^2(25^\circ) + \cos^2(65^\circ)\)
- \(\cos^2(30^\circ) + \cos^2(60^\circ)\)
- \(\cos^2(35^\circ) + \cos^2(55^\circ)\)
- \(\cos^2(40^\circ) + \cos^2(50^\circ)\)

### Step 2: Use the identity
Using the identity \(\cos^2 \theta + \cos^2 (90^\circ - \theta) = 1\), we can simplify each pair:
- \(\cos^2(5^\circ) + \cos^2(85^\circ) = \cos^2(5^\circ) + \sin^2(5^\circ) = 1\)
- \(\cos^2(10^\circ) + \cos^2(80^\circ) = 1\)
- \(\cos^2(15^\circ) + \cos^2(75^\circ) = 1\)
- \(\cos^2(20^\circ) + \cos^2(70^\circ) = 1\)
- \(\cos^2(25^\circ) + \cos^2(65^\circ) = 1\)
- \(\cos^2(30^\circ) + \cos^2(60^\circ) = 1\)
- \(\cos^2(35^\circ) + \cos^2(55^\circ) = 1\)
- \(\cos^2(40^\circ) + \cos^2(50^\circ) = 1\)

### Step 3: Count the pairs
There are 8 pairs, each contributing a value of 1 to the sum:
\[ 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 = 8 \]

### Step 4: Add the remaining terms
Now, we need to consider the remaining terms:
- \(\cos^2(45^\circ) = \left(\frac{1}{\sqrt{2}}\right)^2 = \frac{1}{2}\)
- \(\cos^2(90^\circ) = 0\)

### Step 5: Combine all contributions
Adding these contributions together:
\[ 8 + \frac{1}{2} + 0 = 8 + 0.5 = 8.5 \]

### Final Result
Thus, the final answer is:
\[ \frac{17}{2} \]