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Find the minimum value of the expression \\(\left(\sqrt{2(1+\cos 2 x)}-\sqrt{36-4 \sqrt{5}} \sin x+2\right) \cdot \left(3+2 \sqrt{10-\sqrt{5}} \cos y-\cos 2 y\right)\\). If the answer is not an integer, round it to the nearest integer.
First, create a high level overview in <plan></plan> tags, then do your reasoning in <think></think> tags, and put your final answer in <answer></answer>.<plan>
1. Simplify the expression by breaking it down into manageable parts.
2. Use trigonometric identities to simplify terms involving trigonometric functions.
3. Analyze the structure of the expression to identify potential optimization strategies.
4. Consider the possibility of minimizing each part of the expression separately.
5. Calculate the minimum value of the entire expression and round it to the nearest integer if necessary.
</plan>
<think>
Hmm... let me start by simplifying the expression. The expression is:
\[
\left(\sqrt{2(1+\cos 2x)} - \sqrt{36-4\sqrt{5}} \sin x + 2\right) \cdot \left(3 + 2\sqrt{10-\sqrt{5}} \cos y - \cos 2y\right)
\]
First, I notice that \(\sqrt{2(1+\cos 2x)}\) can be simplified using a trigonometric identity. Recall that \(1 + \cos 2x = 2\cos^2 x\), so:
\[
\sqrt{2(1+\cos 2x)} = \sqrt{2 \cdot 2\cos^2 x} = \sqrt{4\cos^2 x} = 2|\cos x|
\]
Since \(\cos x\) can be positive or negative, we'll keep it as \(2|\cos x|\) for now.
Next, let's look at the second part of the first factor: \(-\sqrt{36-4\sqrt{5}} \sin x + 2\). This seems a bit complicated, but perhaps we can find a way to simplify or bound it.
Now, let's consider the second factor: \(3 + 2\sqrt{10-\sqrt{5}} \cos y - \cos 2y\). Again, we can use trigonometric identities to simplify \(\cos 2y\). Recall that \(\cos 2y = 2\cos^2 y - 1\), so:
\[
3 + 2\sqrt{10-\sqrt{5}} \cos y - \cos 2y = 3 + 2\sqrt{10-\sqrt{5}} \cos y - (2\cos^2 y - 1) = 4 + 2\sqrt{10-\sqrt{5}} \cos y - 2\cos^2 y
\]
This is a quadratic in terms of \(\cos y\), so we might be able to find its minimum value.
Wait, perhaps it's better to think about minimizing each factor separately and then multiply the minima together. Let's try that.
First, let's find the minimum of the first factor:
\[
F_1(x) = 2|\cos x| - \sqrt{36-4\sqrt{5}} \sin x + 2
\]
This seems complicated due to the absolute value and the trigonometric functions. Maybe we can express it in terms of a single trigonometric function.
Alternatively, perhaps we can use calculus to find the minimum of \(F_1(x)\). Let's try that.
Let's consider \(F_1(x) = 2|\cos x| - \sqrt{36-4\sqrt{5}} \sin x + 2\). To find the minimum, we can take the derivative and set it to zero.
However, the absolute value makes this tricky. Perhaps we can consider the cases where \(\cos x\) is positive and negative separately.
Case 1: \(\cos x \geq 0\)
Then \(F_1(x) = 2\cos x - \sqrt{36-4\sqrt{5}} \sin x + 2\). Taking the derivative with respect to \(x\):
\[
F_1'(x) = -2\sin x - \sqrt{36-4\sqrt{5}} \cos x
\]
Setting \(F_1'(x) = 0\):
\[
-2\sin x - \sqrt{36-4\sqrt{5}} \cos x = 0 \implies -2\sin x = \sqrt{36-4\sqrt{5}} \cos x \implies \tan x = -\frac{\sqrt{36-4\sqrt{5}}}{2}
\]
This gives us a critical point, but we need to check if it's a minimum.
Alternatively, perhaps we can use trigonometric identities to express \(F_1(x)\) as a single trigonometric function.
Recall that \(A\sin x + B\cos x = R\sin(x + \alpha)\), where \(R = \sqrt{A^2 + B^2}\) and \(\tan \alpha = \frac{B}{A}\).
In our case, \(A = -\sqrt{36-4\sqrt{5}}\) and \(B = 2\), so:
\[
R = \sqrt{(-\sqrt{36-4\sqrt{5}})^2 + 2^2} = \sqrt{36-4\sqrt{5} + 4} = \sqrt{40-4\sqrt{5}} = 2\sqrt{10-\sqrt{5}}
\]
So, \(F_1(x) = 2\sqrt{10-\sqrt{5}} \sin(x + \alpha) + 2\).
The minimum value of \(\sin(x + \alpha)\) is \(-1\), so the minimum of \(F_1(x)\) is:
\[
2\sqrt{10-\sqrt{5}} \cdot (-1) + 2 = -2\sqrt{10-\sqrt{5}} + 2
\]
Now, let's find the minimum of the second factor:
\[
F_2(y) = 4 + 2\sqrt{10-\sqrt{5}} \cos y - 2\cos^2 y
\]
This is a quadratic in \(\cos y\). Let \(z = \cos y\), then:
\[
F_2(y) = -2z^2 + 2\sqrt{10-\sqrt{5}} z + 4
\]
This is a quadratic equation in \(z\), and since the coefficient of \(z^2\) is negative, it has a maximum value. However, we need the minimum value of \(F_2(y)\). The minimum will occur at the boundaries of \(z\), i.e., when \(z = 1\) or \(z = -1\).
Let's evaluate \(F_2(y)\) at \(z = 1\) and \(z = -1\).
At \(z = 1\):
\[
F_2(y) = -2(1)^2 + 2\sqrt{10-\sqrt{5}} (1) + 4 = -2 + 2\sqrt{10-\sqrt{5}} + 4 = 2 + 2\sqrt{10-\sqrt{5}}
\]
At \(z = -1\):
\[
F_2(y) = -2(-1)^2 + 2\sqrt{10-\sqrt{5}} (-1) + 4 = -2 - 2\sqrt{10-\sqrt{5}} + 4 = 2 - 2\sqrt{10-\sqrt{5}}
\]
Since \(\sqrt{10-\sqrt{5}} > 0\), the minimum value of \(F_2(y)\) is \(2 - 2\sqrt{10-\sqrt{5}}\).
Now, the minimum value of the entire expression is the product of the minima of \(F_1(x)\) and \(F_2(y)\):
\[
(-2\sqrt{10-\sqrt{5}} + 2) \cdot (2 - 2\sqrt{10-\sqrt{5}}) = (-2\sqrt{10-\sqrt{5}} + 2)(2 - 2\sqrt{10-\sqrt{5}})
\]
Let's compute this product:
\[
(-2\sqrt{10-\sqrt{5}} + 2)(2 - 2\sqrt{10-\sqrt{5}}) = (-2\sqrt{10-\sqrt{5}} \cdot 2) + (-2\sqrt{10-\sqrt{5}} \cdot -2\sqrt{10-\sqrt{5}}) + (2 \cdot 2) + (2 \cdot -2\sqrt{10-\sqrt{5}})
\]
Simplifying each term:
\[
= -4\sqrt{10-\sqrt{5}} + 4(10-\sqrt{5}) + 4 - 4\sqrt{10-\sqrt{5}}
\]
Combine like terms:
\[
= -4\sqrt{10-\sqrt{5}} - 4\sqrt{10-\sqrt{5}} + 4(10-\sqrt{5}) + 4 = -8\sqrt{10-\sqrt{5}} + 40 - 4\sqrt{5} + 4 = -8\sqrt{10-\sqrt{5}} + 44 - 4\sqrt{5}
\]
This expression is quite complicated, and calculating its exact value might not be straightforward. However, we can approximate the value.
First, let's approximate \(\sqrt{10 - \sqrt{5}}\). Since \(\sqrt{5} \approx 2.236\), we have:
\[
\sqrt{10 - 2.236} = \sqrt{7.764} \approx 2.786
\]
Similarly, \(\sqrt{5} \approx 2.236\).
Now, let's approximate the expression:
\[
-8 \cdot 2.786 + 44 - 4 \cdot 2.236 = -22.288 + 44 - 8.944 = 12.768
\]
Rounding this to the nearest integer gives us 13.
Wait, let's double-check the calculations to ensure accuracy.
First, let's compute \(\sqrt{10 - \sqrt{5}}\) more precisely.
\(\sqrt{5} \approx 2.2360679775\), so:
\[
\sqrt{10 - 2.2360679775} = \sqrt{7.7639320225} \approx 2.786
\]
Now, compute the expression:
\[
-8 \cdot 2.786 + 44 - 4 \cdot 2.236 = -22.288 + 44 - 8.944 = 12.768
\]
Rounding 12.768 to the nearest integer gives us 13.
Thus, the minimum value of the expression is approximately 13.
</think>
<answer>
\boxed{13}
</answer>