Finding value of determinant, Where $A+B+C+P+Q+R=0$ Finding value of Determinant $$\begin{vmatrix}
\tan (A+P) &amp; \tan(B+P)  &amp; \tan(C+P)\\\\ 
 \tan (A+Q)&amp; \tan (B+Q)  &amp; \tan (C+Q)\\\\ 
 \tan (A+R)&amp;\tan (B+R)  &amp; \tan(C+R) 
\end{vmatrix}$$ for all values of $A,B,C,P,Q,R$ Where $A+B+C+P+Q+R=0$

I did not understand how to start it, Could some help me, Thanks
We have that if $X+Y+Z = 0$, then $\tan X+ \tan Y+\tan Z = \tan X \tan Y \tan Z$.

Hence for example $$\tan (A+P) \tan (B+Q) \tan (C+R) = \tan (A+P) +\tan (B+Q) +\tan (C+R)$$

Hence when you expand from R1, the determinant expands as 

$$\begin{align} &amp; \left(\tan (B+Q) +\tan (C+R)-\tan (C+Q) - \tan (C+R)\right)\\
+ &amp; \left(\tan (C+Q)+\tan (A+R) - \tan (A+Q)-\tan (C+R)\right)\\
+ &amp; \left(\tan (A+Q)+\tan (B+R) - \tan (A+R) - \tan (B+Q)\right)\end{align} = 0$$
We want to find the determinant of this Matrix.
$$
\begin{pmatrix}
Tan(A+P) &amp; Tan(B+P) &amp; Tan(C+P)\\
Tan(A+Q) &amp; Tan(B+Q) &amp; Tan(C+Q)\\
Tan(A+R) &amp; Tan(B+R) &amp; Tan(C+R)
\end{pmatrix}
$$
The determinant of this matrix gives us \to
$$
-Tan(A+R) \cdot Tan(B+Q) \cdot Tan(C+P)\\
+Tan(A+Q) \cdot Tan(B+R) \cdot Tan(C+P)\\
+Tan(A+R) \cdot Tan(B+P) \cdot Tan(C+Q)\\
-Tan(A+P) \cdot Tan(B+R) \cdot Tan(C+Q)\\
-Tan(A+Q) \cdot Tan(B+P) \cdot Tan(C+R)\\
+Tan(A+P) \cdot Tan(B+Q) \cdot Tan(C+R)
$$
But I still don't know how to depress it after this, even though you said
$
A+B+C+P+Q+R = 0
$