If `x=gamma sin theta cos phi, y=gamma sin theta sin phi, z =gamma cos theta` , what is the eliminant of `theta` and `phi` ?

Answer

To eliminate the variables \( \theta \) and \( \phi \) from the given equations \( x = \gamma \sin \theta \cos \phi \), \( y = \gamma \sin \theta \sin \phi \), and \( z = \gamma \cos \theta \), we can follow these steps:

### Step 1: Square the equations
First, we will square each of the equations:
\[ x^2 = \gamma^2 \sin^2 \theta \cos^2 \phi \]
\[ y^2 = \gamma^2 \sin^2 \theta \sin^2 \phi \]
\[ z^2 = \gamma^2 \cos^2 \theta \]

### Step 2: Add the squares of \( x \) and \( y \)
Next, we will add the squares of \( x \) and \( y \):
\[ x^2 + y^2 = \gamma^2 \sin^2 \theta \cos^2 \phi + \gamma^2 \sin^2 \theta \sin^2 \phi \]
Factoring out \( \gamma^2 \sin^2 \theta \):
\[ x^2 + y^2 = \gamma^2 \sin^2 \theta (\cos^2 \phi + \sin^2 \phi) \]
Using the Pythagorean identity \( \cos^2 \phi + \sin^2 \phi = 1 \):
\[ x^2 + y^2 = \gamma^2 \sin^2 \theta \]

### Step 3: Substitute \( z^2 \)
Now, we can express \( z^2 \) in terms of \( \theta \):
\[ z^2 = \gamma^2 \cos^2 \theta \]

### Step 4: Combine the results
Now we have:
\[ x^2 + y^2 = \gamma^2 \sin^2 \theta \]
\[ z^2 = \gamma^2 \cos^2 \theta \]
We can add these two equations:
\[ x^2 + y^2 + z^2 = \gamma^2 \sin^2 \theta + \gamma^2 \cos^2 \theta \]
Factoring out \( \gamma^2 \):
\[ x^2 + y^2 + z^2 = \gamma^2 (\sin^2 \theta + \cos^2 \theta) \]
Using the Pythagorean identity \( \sin^2 \theta + \cos^2 \theta = 1 \):
\[ x^2 + y^2 + z^2 = \gamma^2 \]

### Final Result
Thus, the eliminant of \( \theta \) and \( \phi \) is:
\[ x^2 + y^2 + z^2 = \gamma^2 \]