`a^2+b^2+2a b+2b c+2c a`

Prove that: 2a^2+2b^2+2c^2-2a b-2b c-2c a=[(a-b)^2+(b-c)^2+(c-a)^2]

Prove that: 2a^2+2b^2+2c^2-2a b-2b c-2c a=[(a-b)^2+(b-c)^2+(c-a)^2]

Prove that: 2a^2+2b^2+2c^2-2a b-2b c-2c a=(a-b)^2+(b-c)^2+(c-a)^2

Compute the following: (i) [a b-b a]+[a bb a] (ii) [a^2+b^2b^2+c^2a^2+c^2a^2+b^2]+[2a b2b c-2a c-2a b] (iii) [-1 4-6 8 5 16 2 8 5]+[12 7 6 8 0 5 3 2 4] (iv) [cos^2xsin^2xsin^2xcos^2x]+[sin^2xcos^2xcos^2xsin^2x]

Show that: |b^2+c^2a b a c b a c^2+a^2b c c a c b a^2+b^2|=4a^2b^2c^2

Compute the following: i) [[a, b],[ -b, a]]+[[a, b],[ b , a]] . ii) [[a^2+b^2 , b^2+c^2], [a^2+c^2 , a^2+b^2]]+[[2 a b , 2 b c],[ -2 a c, -2 a b]] iii) [[-1 , 4 ,-6] ,[ 8, 5 , 16],[ 2 , 8 , 5]]+[[12, 7 ,6 ],[ 8, 0, 5],[ 3, 2, 4]] iv) [[cos ^2 x ,. sin ^2 x],[ sin ^2 x , cos ^2 x]]+[[sin ^2 x, cos ^2 x ],[ cos ^2 x ,sin ^2 x]] .

Prove that: |[b, c-a^2,c] ,[a-b^2,a b-c^2,c ],[a-b^2,a ,b-c^2b c-a^2a b-c^2b c-a^2c a-b^2]|=|[a, b, c],[ b ,c ,a],[ c, a ,b]|^2 .

In triangle A B C ,2a csin(1/2(A-B+C)) is equal to (a) a^2+b^2-c^2 (b) c^2+a^2-b^2 (c) b^2-c^2-a^2 (d) c^2-a^2-b^2

In any A B C , the value of 2a csin((A-B+C)/2) is (a) a^2+b^2-c^2 (b) c^2+a^2-b^2 (c) b^2-c^2-a^2 (d) c^2-a^2-b^2

In a A B C ,2a csin((A-B+C))/2= (A) a^2+b^2+c^2 (B) c^2+a^2-b^2 (C) b^2-c^2-a^2 (D) c^2-a^2+b^2