Ques.> Show that the value of the determinent vanishes for all values of A, B, C, P, Q and R where A+B+C+P+Q+R=0.

Pls. I want solution.

Let us have.....  D =

Now as it is given,,,,

A+B+C+P+Q+R = 0 rewriting it as  (A+P) + (B+Q) + (180 + C+R)  = 180...

so we can say....(A+P) , (B+Q) , (180 + C+R) are three angles of a triangle....

similarly....rearranging them again we will get more SUCH equations....from which we can ssay...... (A+P) , (B+R) , (180 + C+Q) are three angles of a triangle....

and also....... (A+Q) , (B+P) , (180 + C+R) are three angles of a triangle....

and also....... (A+R) , (B+P) , (180 + C+Q) are three angles of a triangle....

and also....... (A+Q) , (B+R) , (180 + C+P) are three angles of a triangle....

and also....... (A+R) , (B+Q) , (180 + C+P) are three angles of a triangle....

PHEW.....the reason behind doing all this is comin....next.........

lets go back to the determinant....D.....

If we open it from the first row....then.....we will have.........

D = tan(A+P) { tan(B+Q)tan (C+R) - tan(C+Q)tan (B+R) } - tan (B+P) { tan(A+Q)tan(C+R) - tan(C+Q)tan(A+R) } + tan(C+P) { tan(A+Q)tan(B+R) -  tan(B+Q)tan(A+R)}

now the main thing starts.......

when u open these brackets....u will get many terms of the form...

tan X tan Y tan Z........X,Y,Z may take values like-- (A+P), (B+Q) , (C+R) etc....

Now as we know.......in any triangle...say triangle XYZ....

X + Y + Z =180.....and for these angles.......

tanX tanY tanZ = tanX + tanY + tanZ

and using this property we can split those terms of multiplication form into addition.......and finally.....the opposite signs will cancel out.....

AND U WILL GET A BIG ZERO.........

hope its clear buddy.......

Aatish what an aatishi solution. Great effort i should say. well done.