Simplifying the Expression 1 – cos(x1)cos(x)
The expression we want to simplify is 1 – cos(x1)cos(x). To do this, we’ll employ a useful trigonometric identity.
Using the Product-to-Sum Identities
One effective method for simplifying products of cosine functions is to use the product-to-sum identities. Specifically, we can express the product of cosines as:
cos(A)cos(B) = 0.5 [cos(A + B) + cos(A - B)]Applying this identity to our expression where A = x1 and B = x, we get:
cos(x1)cos(x) = 0.5[cos(x1 + x) + cos(x1 - x)]Substituting Back into the Expression
Now, let’s substitute this back into our original expression:
1 - cos(x1)cos(x) = 1 - 0.5[cos(x1 + x) + cos(x1 - x)]This simplifies to:
= 1 - 0.5cos(x1 + x) - 0.5cos(x1 - x)Final Simplified Expression
Thus, the simplified form of the original expression 1 – cos(x1)cos(x) can be expressed as:
1 - 0.5cos(x1 + x) - 0.5cos(x1 - x)This simplification shows how we can break down the product of cosines into a more manageable form using trigonometric identities.