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. To write the expression cos(5x)cos(-4x) as a sum or difference, we can use the cosine double angle formula, which states that cos(a)cos(b) = (1/2)(cos(a+b) + cos(a-b)).
Using this formula, we can rewrite cos(5x)cos(-4x) as:
(1/2)(cos(5x+(-4x)) + cos(5x-(-4x)))
Simplifying this expression further:
(1/2)(cos(x) + cos(9x))
Therefore, cos(5x)cos(-4x) can be written as the sum of cos(x) and cos(9x).
2. To write the expression cos(4x)-cos(12x) as a product, we can use the cosine addition formula, which states that cos(a) - cos(b) = -2sin((a+b)/2)sin((a-b)/2).
Using this formula, we can rewrite cos(4x) - cos(12x) as:
-2sin((4x+12x)/2)sin((4x-12x)/2)
Simplifying this expression further:
-2sin(8x)sin(-4x)
Since sin(-x) = -sin(x), we can rewrite the expression as:
2sin(8x)sin(4x)
Therefore, cos(4x) - cos(12x) can be written as the product of 2sin(8x) and sin(4x).
To write the expression cos(5x)cos(-4x) as a sum or difference, you can use the identity:
cos(a)cos(b) = (1/2)[cos(a+b) + cos(a-b)]
Applying this identity to the given expression, we have:
cos(5x)cos(-4x) = (1/2)[cos(5x + (-4x)) + cos(5x - (-4x))]
Simplifying further, we have:
cos(5x)cos(-4x) = (1/2)[cos(x) + cos(9x)]
Therefore, cos(5x)cos(-4x) can be written as the sum of cosine functions: cos(x) + cos(9x).
Now, let's move on to the second question:
To write the expression cos(4x)-cos(12x) as a product, we can use the identity:
cos(a) - cos(b) = -2sin[(a+b)/2]sin[(a-b)/2]
Applying this identity to the given expression, we have:
cos(4x) - cos(12x) = -2sin[(4x + 12x)/2]sin[(4x - 12x)/2]
Simplifying further, we have:
cos(4x) - cos(12x) = -2sin[8x/2]sin[-8x/2]
cos(4x) - cos(12x) = -2sin[4x]sin[-4x]
Therefore, cos(4x) - cos(12x) can be written as the product of two sine functions: -2sin(4x)sin(-4x).