Prove that `(cos A - cos B) ^(2) + (sin A - sin B ) ^(2) = 4 sin ^(2) ((A -B)/( 2 )).`

Prove that (cos A + cos B ) ^(2) +(sin A + sin B) ^(2) = 4 cos ^(2) ((A -B)/(2)).

Prove that (cosA-cosB)^2+(sin A-sin B)^2=4sin^2((A-B)/2)

Prove that ((cos A + cos B)/( sin A - sin B )) ^(n) + ((sin A + sin B )/( cos A - cos B )) ^(n) ={{:(2 cot ^(n) ((A- B)/(2)),"," "if n is even."),(0,",""if n is odd."):}.

Prove that : cot^(2) A - cot^(2) B = (cos^(2) A - cos^(2) B)/ (sin^(2) A sin^(2) B) = cosec^(2) A - cosec^(2)B

Prove that a cos A + b cos B + c cos C = 4 R sin A sin B sin C.

(cos 2B - cos 2 A)/( sin 2 A + sin 2 B) = tan (A -B).

Prove that : 2 sin^(2) A + cos^(4) A = 1 + sin^(4) A

Prove that : (sin A - sin B)/ (cos A + cos B) + (cos A - cos B)/ (sin A + sin B) = 0

If A and B are complementary angles, prove that : (sin A + sin B)/ (sin A - sin B) + (cos B - cos A)/ (cos B + cos A) = (2)/(2 sin^(2) A - 1)

If A+B+C=180^0 , prove that : cos^2( A/2) + cos^2( B/2) + cos^2(C/2) = 2+2 sin(A/2) sin( B/2) sin( C/2)