The given expression cos(5x)cos(3x)−sin(5x)sin(3x) simplifies to cos(2x), by applying the cosine of a difference trigonometric identity.
The expression cos(5x)cos(3x)−sin(5x)sin(3x) can be simplified using a trigonometric identity known as the cosine of a sum or difference. Specifically, the identity is cos(a − b) = cos(a)cos(b) + sin(a)sin(b). By applying this identity to the given expression, we get:
cos(5x − 3x) = cos(5x)cos(3x) + sin(5x)sin(3x)
However, the given expression is actually the negative of the sin terms, which aligns with the cosine of a difference:
cos(5x)cos(3x) − sin(5x)sin(3x) = cos(5x − 3x)
Therefore, the original expression simplifies to:
cos(2x)