1.Write the expression as a sum or difference.
cos(5x)cos(-4x)
2.Write the expression as a product.
cos(4x)-cos(12x)
1. use
cos(a)+cos(b)=2[cos((a+b)/2)cos((a-b)/2)]
and set a=x/2, b=9x/2
2. use
cos(a)-cos(b)=-2[sin((a+b)/2)sin((a-b)/2)]
and set a=-4x, b=8x
Hint: before doing the work, check the formulas using some trivial cases such as a=π/2, and b=π/4.
To write the expression as a sum or difference, we can use the trigonometric identity:
cos(A)cos(B) = 1/2 * [cos(A + B) + cos(A - B)]
For the expression cos(5x)cos(-4x), we can replace A with 5x and B with -4x:
cos(5x)cos(-4x) = 1/2 * [cos(5x + (-4x)) + cos(5x - (-4x))]
Simplifying further:
= 1/2 * [cos(x) + cos(9x)]
Therefore, cos(5x)cos(-4x) can be written as the sum of cos(x) and cos(9x).
To write the expression cos(4x) - cos(12x) as a product, we can use the trigonometric identity:
cos(A) - cos(B) = -2 * [sin((A+B)/2) * sin((A-B)/2)]
For the expression cos(4x) - cos(12x), we can replace A with 4x and B with 12x:
cos(4x) - cos(12x) = -2 * [sin((4x + 12x)/2) * sin((4x - 12x)/2)]
Simplifying further:
= -2 * [sin(8x) * sin(-4x)]
Now, we can use another trigonometric identity:
sin(-θ) = -sin(θ)
= -2 * [-sin(8x) * sin(4x)]
= 2 * [sin(8x) * sin(4x)]
Therefore, cos(4x) - cos(12x) can be written as the product of sin(8x) and sin(4x).