If `(2tan^(2)theta_(1)tan^(2)theta_(2)tan^(2)theta_(3)+tan^(2)theta_(1)tan^(2)theta_(2)+tan^(2)theta_(2)tan^(2)theta_(3)+tan^(2)theta_(3)tan^(2)theta_(1)=1` then which of the following relations hold good ?

Options

A. (a)`sin^(2)theta_(1) + sin^(2)theta_(2) + sin^(2)theta_(3) = 1`
B. (b)`cos2theta_(1) + cos2theta_(2) + cos2theta_(3) = 1`
C. (c)`sin^(2)theta_(1) + sin^(2)theta_(2) + sin^(2)theta_(3) = 2`
D. (d)`cos2theta_(1) + cos2theta_(2) + cos2theta_(3) = -1`

Explanation

To solve the equation

\[ 2\tan^2\theta_1\tan^2\theta_2\tan^2\theta_3 + \tan^2\theta_1\tan^2\theta_2 + \tan^2\theta_2\tan^2\theta_3 + \tan^2\theta_3\tan^2\theta_1 = 1, \]

we will follow these steps:

### Step 1: Divide the entire equation by \(\tan^2\theta_1\tan^2\theta_2\tan^2\theta_3\)

We start by dividing each term by \(\tan^2\theta_1\tan^2\theta_2\tan^2\theta_3\):

\[ \frac{2\tan^2\theta_1\tan^2\theta_2\tan^2\theta_3}{\tan^2\theta_1\tan^2\theta_2\tan^2\theta_3} + \frac{\tan^2\theta_1\tan^2\theta_2}{\tan^2\theta_1\tan^2\theta_2\tan^2\theta_3} + \frac{\tan^2\theta_2\tan^2\theta_3}{\tan^2\theta_1\tan^2\theta_2\tan^2\theta_3} + \frac{\tan^2\theta_3\tan^2\theta_1}{\tan^2\theta_1\tan^2\theta_2\tan^2\theta_3} = \frac{1}{\tan^2\theta_1\tan^2\theta_2\tan^2\theta_3} \]

This simplifies to:

\[ 2 + \cot^2\theta_1 + \cot^2\theta_2 + \cot^2\theta_3 = \cot^2\theta_1 \cot^2\theta_2 \cot^2\theta_3. \]

### Step 2: Substitute \(\cot^2\theta_i\) with \(\cot^2\theta_i - 1\)

Using the identity \(\cot^2\theta = \cot^2\theta - 1 + 1\), we can rewrite the equation as:

\[ 2 + (\cot^2\theta_1 - 1) + (\cot^2\theta_2 - 1) + (\cot^2\theta_3 - 1) = (\cot^2\theta_1 - 1)(\cot^2\theta_2 - 1)(\cot^2\theta_3 - 1). \]

This simplifies to:

\[ 2 + \cot^2\theta_1 + \cot^2\theta_2 + \cot^2\theta_3 - 3 = (\cot^2\theta_1 - 1)(\cot^2\theta_2 - 1)(\cot^2\theta_3 - 1). \]

### Step 3: Simplify the left-hand side

The left-hand side becomes:

\[ \cot^2\theta_1 + \cot^2\theta_2 + \cot^2\theta_3 - 1. \]

### Step 4: Expand the right-hand side

The right-hand side expands to:

\[ \cot^2\theta_1 \cot^2\theta_2 \cot^2\theta_3 - (\cot^2\theta_1 + \cot^2\theta_2 + \cot^2\theta_3) + 1. \]

### Step 5: Set the equation

Now we have:

\[ \cot^2\theta_1 + \cot^2\theta_2 + \cot^2\theta_3 - 1 = \cot^2\theta_1 \cot^2\theta_2 \cot^2\theta_3 - (\cot^2\theta_1 + \cot^2\theta_2 + \cot^2\theta_3) + 1. \]

### Step 6: Rearranging terms

Rearranging gives:

\[ 2(\cot^2\theta_1 + \cot^2\theta_2 + \cot^2\theta_3) = \cot^2\theta_1 \cot^2\theta_2 \cot^2\theta_3 + 2. \]

### Step 7: Final conclusion

This leads us to the relation:

\[ \cot^2\theta_1 + \cot^2\theta_2 + \cot^2\theta_3 = 1. \]

Thus, the relations that hold good are:

- \( \sin^2\theta_1 + \sin^2\theta_2 + \sin^2\theta_3 = 1 \)
- \( \cos^2\theta_1 + \cos^2\theta_2 + \cos^2\theta_3 = 1 \)