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

Prove that (cos A - cos B) ^(2) + (sin A - sin B ) ^(2) = 4 sin ^(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 a cos A + b cos B + c cos C = 4 R sin A sin B sin C.

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)

cos (A +B) + sin (A -B) = 2 sin (45 ^(@) + A) cos ( 45 ^(@) + B).

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

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

If A + B + C = pi then prove that cos A + cos B + cos C = 1 + 4 sin(A/2) .sin(B/2).sin(C/2)

Prove the following identities : (cos A + sin A)^(2) + (cos A - sin A)^(2) = 2

Prove that: (sin A-sin B)/(cos A+cos B)=tan((A-B)/2)