If `( sec theta + tan theta ) ( sec phi + tan phi ) ( sec Psi + tan Psi ) = tan theta . tan phi . tan Psi `, then `( sec theta - tan theta ) ( sec phi - tan phi ) ( sec Psi - tan Psi ) =`

Options

A. `cot theta cot phi cot Psi`
B. `tan theta tan phi tan Psi`
C. `tan theta + tan phi + tan Psi`
D. `cot theta + cot phi + cot Psi`

Answer

To solve the equation \(( \sec \theta + \tan \theta )( \sec \phi + \tan \phi )( \sec \psi + \tan \psi ) = \tan \theta \tan \phi \tan \psi\), we need to find the value of \(( \sec \theta - \tan \theta )( \sec \phi - \tan \phi )( \sec \psi - \tan \psi )\).

### Step 1: Recall the identity
We start by recalling the trigonometric identity:
\[ \sec^2 x - \tan^2 x = 1 \]
This can be factored as:
\[ (\sec x - \tan x)(\sec x + \tan x) = 1 \]

### Step 2: Express \(\sec x - \tan x\)
From the identity, we can express \(\sec x - \tan x\) as:
\[ \sec x - \tan x = \frac{1}{\sec x + \tan x} \]

### Step 3: Apply the identity to each term
Using the identity for each angle:
\[ \sec \theta - \tan \theta = \frac{1}{\sec \theta + \tan \theta} \]
\[ \sec \phi - \tan \phi = \frac{1}{\sec \phi + \tan \phi} \]
\[ \sec \psi - \tan \psi = \frac{1}{\sec \psi + \tan \psi} \]

### Step 4: Substitute into the expression
Now we substitute these into the expression we want to find:
\[ (\sec \theta - \tan \theta)(\sec \phi - \tan \phi)(\sec \psi - \tan \psi) = \left(\frac{1}{\sec \theta + \tan \theta}\right)\left(\frac{1}{\sec \phi + \tan \phi}\right)\left(\frac{1}{\sec \psi + \tan \psi}\right) \]

### Step 5: Combine the fractions
This simplifies to:
\[ \frac{1}{(\sec \theta + \tan \theta)(\sec \phi + \tan \phi)(\sec \psi + \tan \psi)} \]

### Step 6: Substitute the given equation
From the given equation, we know:
\[ (\sec \theta + \tan \theta)(\sec \phi + \tan \phi)(\sec \psi + \tan \psi) = \tan \theta \tan \phi \tan \psi \]
Thus, we can substitute this into our expression:
\[ \frac{1}{\tan \theta \tan \phi \tan \psi} \]

### Step 7: Convert to cotangent
We can express this as:
\[ \frac{1}{\tan \theta} \cdot \frac{1}{\tan \phi} \cdot \frac{1}{\tan \psi} = \cot \theta \cdot \cot \phi \cdot \cot \psi \]

### Final Result
Therefore, the value of \(( \sec \theta - \tan \theta )( \sec \phi - \tan \phi )( \sec \psi - \tan \psi )\) is:
\[ \cot \theta \cdot \cot \phi \cdot \cot \psi \]