If `x+y+z=0, |x|=|y|=|z|=2 and theta` is angle between y and z, then the value of `cosec^(2)theta+cot^(2)theta` is equal to

Options

A. `(4)/(3)`
B. `(5)/(3)`
C. `(1)/(3)`
D. `1`

Answer

To solve the problem step by step, we will follow the reasoning outlined in the video transcript.

### Step 1: Understand the given conditions
We have the equations:
1. \( x + y + z = 0 \)
2. \( |x| = |y| = |z| = 2 \)

From the second condition, we know that the magnitudes of vectors \( x, y, z \) are equal to 2.

### Step 2: Express one vector in terms of the others
From \( x + y + z = 0 \), we can express \( x \) as:
\[ x = - (y + z) \]

### Step 3: Calculate the dot products
We will take the dot product of the vectors with themselves:
1. \( x \cdot x = |x|^2 = 4 \)
2. \( y \cdot y = |y|^2 = 4 \)
3. \( z \cdot z = |z|^2 = 4 \)

Now, we can use the equation \( x + y + z = 0 \) to derive relationships between the dot products:
- Taking the dot product of \( x \) with the equation \( x + y + z = 0 \):
\[ x \cdot x + x \cdot y + x \cdot z = 0 \]
This gives:
\[ 4 + x \cdot y + x \cdot z = 0 \]
Thus:
\[ x \cdot y + x \cdot z = -4 \]

- Similarly, taking the dot product with \( y \):
\[ y \cdot x + y \cdot y + y \cdot z = 0 \]
This gives:
\[ y \cdot x + 4 + y \cdot z = 0 \]
Thus:
\[ y \cdot x + y \cdot z = -4 \]

- And with \( z \):
\[ z \cdot x + z \cdot y + z \cdot z = 0 \]
This gives:
\[ z \cdot x + z \cdot y + 4 = 0 \]
Thus:
\[ z \cdot x + z \cdot y = -4 \]

### Step 4: Relate the dot products
From the three equations derived, we can conclude:
\[ x \cdot y = x \cdot z = y \cdot z \]

Let \( y \cdot z = k \). Then we have:
1. \( k + 4 + k = 0 \) (from \( x \cdot y + x \cdot z = -4 \))
2. This simplifies to \( 2k + 4 = 0 \) leading to \( k = -2 \).

Thus:
\[ y \cdot z = -2 \]

### Step 5: Calculate \( \cos \theta \)
Using the formula for the cosine of the angle between two vectors:
\[ \cos \theta = \frac{y \cdot z}{|y| |z|} \]
Substituting the values:
\[ \cos \theta = \frac{-2}{2 \cdot 2} = \frac{-2}{4} = -\frac{1}{2} \]

### Step 6: Find \( \theta \)
The angle \( \theta \) for which \( \cos \theta = -\frac{1}{2} \) is:
\[ \theta = \frac{2\pi}{3} \]

### Step 7: Calculate \( \csc^2 \theta + \cot^2 \theta \)
1. First, calculate \( \csc^2 \theta \):
\[ \csc \theta = \frac{1}{\sin \theta} \]
Since \( \sin \theta = \sqrt{1 - \cos^2 \theta} = \sqrt{1 - \left(-\frac{1}{2}\right)^2} = \sqrt{1 - \frac{1}{4}} = \sqrt{\frac{3}{4}} = \frac{\sqrt{3}}{2} \)
Thus,
\[ \csc \theta = \frac{2}{\sqrt{3}} \]
So,
\[ \csc^2 \theta = \left(\frac{2}{\sqrt{3}}\right)^2 = \frac{4}{3} \]

2. Now calculate \( \cot^2 \theta \):
\[ \cot \theta = \frac{\cos \theta}{\sin \theta} = \frac{-\frac{1}{2}}{\frac{\sqrt{3}}{2}} = -\frac{1}{\sqrt{3}} \]
Thus,
\[ \cot^2 \theta = \left(-\frac{1}{\sqrt{3}}\right)^2 = \frac{1}{3} \]

### Step 8: Combine the results
Now we can find:
\[ \csc^2 \theta + \cot^2 \theta = \frac{4}{3} + \frac{1}{3} = \frac{5}{3} \]

### Final Answer
The value of \( \csc^2 \theta + \cot^2 \theta \) is:
\[ \frac{5}{3} \]

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