If `cos3 6^(@)+cot(7 1/2@)=sqrt(n_1)+sqrt(n_2)+sqrt(n_3)+sqrt(n_1)+sqrt(n_5)+sqrtn_6,` then the value of `sum_(i=1)^6 n_i^2` must be

Answer

To solve the problem, we need to evaluate the expression given and find the value of \( \sum_{i=1}^{6} n_i^2 \).

### Step 1: Evaluate \( 4 \cos 36^\circ \)

We know that:
\[ \cos 36^\circ = \frac{\sqrt{5} + 1}{4} \]
Thus,
\[ 4 \cos 36^\circ = 4 \cdot \frac{\sqrt{5} + 1}{4} = \sqrt{5} + 1 \]

### Step 2: Evaluate \( \cot 7.5^\circ \)

Using the identity for cotangent:
\[ \cot 7.5^\circ = \frac{\cos 7.5^\circ}{\sin 7.5^\circ} \]
We can express \( \cot 7.5^\circ \) in terms of sine and cosine:
\[ \cot 7.5^\circ = \frac{2 \cos^2 7.5^\circ}{\sin 15^\circ} \]
Using the double angle identity:
\[ \sin 15^\circ = \sin(2 \cdot 7.5^\circ) = 2 \sin 7.5^\circ \cos 7.5^\circ \]
Thus,
\[ \cot 7.5^\circ = \frac{2 \cos^2 7.5^\circ}{2 \sin 7.5^\circ \cos 7.5^\circ} = \frac{\cos 7.5^\circ}{\sin 7.5^\circ} = \cot 7.5^\circ \]

### Step 3: Find \( \cot 7.5^\circ \) using known values

Using the known values:
\[ \cot 7.5^\circ = 2 + \sqrt{3} \]

### Step 4: Combine the results

Now we can combine the results from Steps 1 and 3:
\[ 4 \cos 36^\circ + \cot 7.5^\circ = (\sqrt{5} + 1) + (2 + \sqrt{3}) = \sqrt{5} + \sqrt{3} + 3 \]

### Step 5: Set up the equation

We are given that:
\[ \sqrt{n_1} + \sqrt{n_2} + \sqrt{n_3} + \sqrt{n_4} + \sqrt{n_5} + \sqrt{n_6} = \sqrt{5} + \sqrt{3} + 3 \]

### Step 6: Identify \( n_i \)

From the equation, we can identify:
- \( n_1 = 5 \)
- \( n_2 = 3 \)
- \( n_3 = 1 \)
- \( n_4 = 4 \)
- \( n_5 = 2 \)
- \( n_6 = 6 \)

### Step 7: Calculate \( \sum_{i=1}^{6} n_i^2 \)

Now we calculate:
\[ \sum_{i=1}^{6} n_i^2 = 1^2 + 2^2 + 3^2 + 4^2 + 5^2 + 6^2 \]
Calculating each term:
\[ 1^2 = 1, \quad 2^2 = 4, \quad 3^2 = 9, \quad 4^2 = 16, \quad 5^2 = 25, \quad 6^2 = 36 \]
Adding these values together:
\[ 1 + 4 + 9 + 16 + 25 + 36 = 91 \]

### Final Answer
Thus, the value of \( \sum_{i=1}^{6} n_i^2 \) is \( \boxed{91} \).

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