If `A,B,C,P,Q,R in R` and
`Delta=|(cos(A+P),cos(A+Q),cos(A+R)),(cos(B+P),cos(B+Q),cos(B+R)),(cos(C+P),cos(C+Q),cos(C+R))|`

Options

A. `Delta` depends on P,Q,R
B. `Delta ` depends on A,B,C
C. `Delta` depends on A,B,C,P,Q,R
D. None of these

Answer

To solve the given determinant problem, we need to evaluate the determinant:

\[ \Delta = \begin{vmatrix} \cos(A+P) & \cos(A+Q) & \cos(A+R) \\ \cos(B+P) & \cos(B+Q) & \cos(B+R) \\ \cos(C+P) & \cos(C+Q) & \cos(C+R) \end{vmatrix} \]

### Step 1: Apply the Cosine Addition Formula
Using the cosine addition formula, we can rewrite each entry of the determinant:

\[ \cos(A + P) = \cos A \cos P - \sin A \sin P \]
\[ \cos(A + Q) = \cos A \cos Q - \sin A \sin Q \]
\[ \cos(A + R) = \cos A \cos R - \sin A \sin R \]

Similarly, we can expand the other rows:

\[ \cos(B + P) = \cos B \cos P - \sin B \sin P \]
\[ \cos(B + Q) = \cos B \cos Q - \sin B \sin Q \]
\[ \cos(B + R) = \cos B \cos R - \sin B \sin R \]

\[ \cos(C + P) = \cos C \cos P - \sin C \sin P \]
\[ \cos(C + Q) = \cos C \cos Q - \sin C \sin Q \]
\[ \cos(C + R) = \cos C \cos R - \sin C \sin R \]

### Step 2: Substitute into the Determinant
Now, substituting these expressions back into the determinant, we have:

\[ \Delta = \begin{vmatrix} \cos A \cos P - \sin A \sin P & \cos A \cos Q - \sin A \sin Q & \cos A \cos R - \sin A \sin R \\ \cos B \cos P - \sin B \sin P & \cos B \cos Q - \sin B \sin Q & \cos B \cos R - \sin B \sin R \\ \cos C \cos P - \sin C \sin P & \cos C \cos Q - \sin C \sin Q & \cos C \cos R - \sin C \sin R \end{vmatrix} \]

### Step 3: Factor Out Common Terms
Notice that the first column can be factored out:

\[ \Delta = \begin{vmatrix} \cos P & \cos Q & \cos R \\ \cos P & \cos Q & \cos R \\ \cos P & \cos Q & \cos R \end{vmatrix} - \begin{vmatrix} \sin A \sin P & \sin A \sin Q & \sin A \sin R \\ \sin B \sin P & \sin B \sin Q & \sin B \sin R \\ \sin C \sin P & \sin C \sin Q & \sin C \sin R \end{vmatrix} \]

### Step 4: Evaluate the Determinant
Now, we can see that the determinant has two rows that are identical in the first matrix, which means:

\[ \begin{vmatrix} \cos P & \cos Q & \cos R \\ \cos P & \cos Q & \cos R \\ \cos P & \cos Q & \cos R \end{vmatrix} = 0 \]

And similarly, the second determinant will also evaluate to zero due to the same reasoning (the rows are linearly dependent).

### Conclusion
Thus, we find that:

\[ \Delta = 0 \]

### Final Answer
The determinant does not depend on \( A, B, C, P, Q, R \) since it evaluates to zero regardless of their values. Therefore, the correct option is:

**None of these.**