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 presented in the video transcript.

### Step 1: Write the given condition
We start with the equation:
\[ x + y + z = 0 \]

### Step 2: Multiply by \( x \)
Multiply both sides of the equation by \( x \):
\[ x \cdot x + x \cdot y + x \cdot z = 0 \]
This simplifies to:
\[ |x|^2 + x \cdot y + x \cdot z = 0 \]
Let’s denote this as Equation (1).

### Step 3: Multiply by \( y \)
Now, multiply the original equation by \( y \):
\[ y \cdot x + y \cdot y + y \cdot z = 0 \]
This simplifies to:
\[ x \cdot y + |y|^2 + y \cdot z = 0 \]
Let’s denote this as Equation (2).

### Step 4: Multiply by \( z \)
Next, multiply the original equation by \( z \):
\[ z \cdot x + z \cdot y + z \cdot z = 0 \]
This simplifies to:
\[ x \cdot z + y \cdot z + |z|^2 = 0 \]
Let’s denote this as Equation (3).

### Step 5: Add Equations (2) and (3)
Now, we add Equation (2) and Equation (3):
\[ (x \cdot y + |y|^2 + y \cdot z) + (x \cdot z + y \cdot z + |z|^2) = 0 \]
This gives:
\[ x \cdot y + x \cdot z + |y|^2 + 2(y \cdot z) + |z|^2 = 0 \]

### Step 6: Substitute the magnitudes
Since we know that \( |x| = |y| = |z| = 2 \), we have:
\[ |x|^2 = |y|^2 = |z|^2 = 4 \]
Substituting these values into the equation:
\[ x \cdot y + x \cdot z + 4 + 2(y \cdot z) + 4 = 0 \]
This simplifies to:
\[ x \cdot y + x \cdot z + 2(y \cdot z) + 8 = 0 \]

### Step 7: Rearranging the equation
Rearranging gives us:
\[ x \cdot y + x \cdot z + 2(y \cdot z) = -8 \]

### Step 8: Substitute into Equation (1)
Now substitute the expression for \( x \cdot y + x \cdot z \) from Equation (1):
\[ 4 - (4 + 2(y \cdot z)) = 0 \]
This leads to:
\[ -2(y \cdot z) = -4 \implies y \cdot z = 2 \]

### Step 9: Use the dot product to find \( \cos \theta \)
Using the definition of the dot product:
\[ y \cdot z = |y||z| \cos \theta \]
Substituting the known magnitudes:
\[ 2 = 2 \cdot 2 \cdot \cos \theta \implies \cos \theta = \frac{1}{2} \]
This means:
\[ \theta = \frac{\pi}{3} \]

### Step 10: Calculate \( \csc^2 \theta + \cot^2 \theta \)
We know:
\[ \csc^2 \theta = \frac{1}{\sin^2 \theta}, \quad \cot^2 \theta = \frac{\cos^2 \theta}{\sin^2 \theta} \]
Calculating \( \sin \theta \):
\[ \sin \theta = \frac{\sqrt{3}}{2} \implies \sin^2 \theta = \frac{3}{4} \]
Thus:
\[ \csc^2 \theta = \frac{1}{\frac{3}{4}} = \frac{4}{3} \]
And:
\[ \cot^2 \theta = \frac{\left(\frac{1}{2}\right)^2}{\frac{3}{4}} = \frac{\frac{1}{4}}{\frac{3}{4}} = \frac{1}{3} \]
Now summing these:
\[ \csc^2 \theta + \cot^2 \theta = \frac{4}{3} + \frac{1}{3} = \frac{5}{3} \]

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