To simplify this expression, we will use the sum and difference formulas for sine and cosine.
Recall thatsin(A + B) = sin(A)cos(B) + cos(A)sin(Bsin(A - B) = sin(A)cos(B) - cos(A)sin(Bcos(A + B) = cos(A)cos(B) - sin(A)sin(Bcos(A - B) = cos(A)cos(B) + sin(A)sin(B)
Let's apply these formulas to our expression:
sin(a + π/8) cos(a - π/24= (sin(a)cos(π/8) + cos(a)sin(π/8)) (cos(a)cos(π/24) + sin(a)sin(π/24)= (sin(a)cos(π/8)cos(a)cos(π/24) + sin(a)cos(π/8)sin(a)sin(π/24) + cos(a)sin(π/8)cos(a)cos(π/24) + cos(a)sin(π/8)sin(a)sin(π/24))
Now we can distribute and simplify this expression further.
To simplify this expression, we will use the sum and difference formulas for sine and cosine.
Recall that
sin(A + B) = sin(A)cos(B) + cos(A)sin(B
sin(A - B) = sin(A)cos(B) - cos(A)sin(B
cos(A + B) = cos(A)cos(B) - sin(A)sin(B
cos(A - B) = cos(A)cos(B) + sin(A)sin(B)
Let's apply these formulas to our expression:
sin(a + π/8) cos(a - π/24
= (sin(a)cos(π/8) + cos(a)sin(π/8)) (cos(a)cos(π/24) + sin(a)sin(π/24)
= (sin(a)cos(π/8)cos(a)cos(π/24) + sin(a)cos(π/8)sin(a)sin(π/24) + cos(a)sin(π/8)cos(a)cos(π/24) + cos(a)sin(π/8)sin(a)sin(π/24))
Now we can distribute and simplify this expression further.