`(cos^2 x+1/(cos^2 x))(1+tan^2 2 y)(3+sin 3 z)=4,` then y can take values equal to

Options

A. x is an integral multiple of `pi`
B. x cannot be an even multiple of `pi`
C. z is an integral multiple of `pi`
D. y is an integral multiple of `pi//2`

Explanation

To solve the equation

\[ \left(\cos^2 x + \frac{1}{\cos^2 x}\right)(1 + \tan^2 2y)(3 + \sin 3z) = 4, \]

we will analyze each part of the equation step by step.

### Step 1: Analyze \(\cos^2 x + \frac{1}{\cos^2 x}\)

Using the AM-GM inequality, we know that:

\[ \cos^2 x + \frac{1}{\cos^2 x} \geq 2. \]

The equality holds when \(\cos^2 x = 1\), which implies \(\cos x = \pm 1\). Therefore, the minimum value of \(\cos^2 x + \frac{1}{\cos^2 x}\) is 2.

### Step 2: Analyze \(1 + \tan^2 2y\)

We know that:

\[ 1 + \tan^2 \theta = \sec^2 \theta. \]

Thus, we can write:

\[ 1 + \tan^2 2y = \sec^2 2y \geq 1. \]

The minimum value occurs when \(\tan 2y = 0\), which gives \(2y = n\pi\) for \(n \in \mathbb{Z}\), leading to:

\[ y = \frac{n\pi}{2}. \]

### Step 3: Analyze \(3 + \sin 3z\)

The sine function has a range of \([-1, 1]\), so:

\[ 3 + \sin 3z \geq 2. \]

The minimum value occurs when \(\sin 3z = -1\), which gives:

\[ 3z = \frac{3\pi}{2} + 2k\pi \implies z = \frac{\pi}{2} + \frac{2k\pi}{3} \quad (k \in \mathbb{Z}). \]

### Step 4: Combine the Results

Now we can combine the results. The equation can be rewritten as:

\[ \left(\cos^2 x + \frac{1}{\cos^2 x}\right)(1 + \tan^2 2y)(3 + \sin 3z) = 4. \]

Given the minimum values we found:

- \(\cos^2 x + \frac{1}{\cos^2 x} \geq 2\),
- \(1 + \tan^2 2y \geq 1\),
- \(3 + \sin 3z \geq 2\).

The product of these minimum values is:

\[ 2 \cdot 1 \cdot 2 = 4. \]

This means that for the equality to hold, we must have:

\[ \cos^2 x + \frac{1}{\cos^2 x} = 2, \quad 1 + \tan^2 2y = 1, \quad 3 + \sin 3z = 2. \]

### Step 5: Solve for \(y\)

From \(1 + \tan^2 2y = 1\), we have:

\[ \tan^2 2y = 0 \implies \tan 2y = 0 \implies 2y = n\pi \implies y = \frac{n\pi}{2}. \]

Thus, the values of \(y\) can be expressed as:

\[ y = \frac{n\pi}{2}, \quad n \in \mathbb{Z}. \]

### Final Answer

The values that \(y\) can take are:

\[ y = \frac{n\pi}{2} \quad (n \in \mathbb{Z}). \]