A) sin 3a - sin 9a

Using the trigonometric identity sin(A) - sin(B) = 2 cos((A+B)/2) sin((A-B)/2), we have:

sin 3a - sin 9a = 2 cos((3a+9a)/2) sin((3a-9a)/2)
= 2 cos(6a/2) sin(-6a/2)
= 2 cos(3a) sin(-3a)
= 2 cos(3a) (-sin(3a))
= -2 sin(3a) cos(3a)

Final Answer: -2sin(3a)cos(3a)

B) cos 13x + cos 12x

This expression does not simplify further. Final Answer: cos 13x + cos 12x

C) sin 2a + sin a / cos 2a - cos a = -ctg(a/2)

To simplify the left side of the equation, we use the trigonometric identity sin(A) + sin(B) = 2 sin((A+B)/2) cos((A-B)/2), and cos(A) - cos(B) = -2 sin((A+B)/2) sin((A-B)/2):

sin 2a + sin a / cos 2a - cos a
= 2 sin((2a+a)/2) cos((2a-a)/2) / -2 sin((2a+a)/2) sin((2a-a)/2)
= 2 sin(3a/2) cos(a/2) / 2 sin(3a/2) sin(a/2)
= cos(a/2) / sin(a/2)

Since cotangent is equal to cosine divided by sine, we have: cot(a/2) = cos(a/2) / sin(a/2)

Final Answer: -ctg(a/2)

D) cos a + sin a / cos a - sin a

This expression does not simplify further. Final Answer: cos a + sin a / cos a - sin a