Answer:
The two rules we need to use are:
Sin(a + b) = sin(a)*cos(b) + sin(b)*cos(a)
cos(a + b) = cos(a)*cos(b) - sin(a)*sin(b)
And we also know that:
sin^2(a) + cos^2(a) = 1
To solve the relations, we start with the left side and try to construct the right side.
a) Sin(3*A) = sin (2*A + A) = sin(2*A)*cos(A) + sin(A)*cos(2*A)
sin(A + A)*cos(A) + sin(A)*cos(A + A)
(sin(A)*cos(A) + sin(A)*cos(A))*cos(A) + sin(A)*(cos(A)*cos(A) - sin(A)*sin(A))
sin(A)*cos^2(A) + sin(A)*cos^2(A) + sin(A)*cos^2(A) - sin^3(A)
3*sin(A)*cos^2(A) - sin(A)*sin^2(A)
sin(A)*(3*cos^2(A) - sin^2(A))
Now we can add and subtract 4*sin^3(A)
sin(A)*(3*cos^2(A) - sin^2(A)) + 4*sin^3(A) - 4*sin^3(A)
sin(A)*(3*cos^2(A) + 3*sin^2(A)) - 4*sin^3(A)
sin(A)*3*(cos^2(A) + sin^2(A)) - 4*sin^3(A)
3*sin(A) - 4*sin^3(A)
b) Here we do the same as before:
cos(3*A) = 4*cos^3(A) - 3*cos(A)
We start with:
Cos(2*A + A) = cos(2*A)*cos(A) - sin(2*A)*sin(A)
= cos(A + A)*cos(A) - sin(A + A)*sin(A)
= (cos(A)*cos(A) - sin(A)*sin(A))*cos(A) - ( sin(A)*cos(A) + sin(A)*cos(A))*sin(A)
= (cos^2(A) - sin^2(A))*cos(A) - sin^2(A)*cos(A) - sin^2(A)*cos(A)
= cos^3(A) - 3*sin^2(A)*cos(A)
= cos(A)*(cos^2(A) - 3*sin^2(A))
now we subtract and add 4*cos^3(A)
= cos(A)*(cos^2(A) - 3*sin^2(A)) + 4*cos^3(A) - 4*cos^3(A)
= cos(A)*(-3*cos^2(A) - 3*sin^2(A)) + 4*cos^3(A)
= cos(A)*(-3)*(cos^2(A) + sin^2(A)) + 4*cos^3(A)
= -3*cos(A) + 4*cos^3(A)