Final answer:
The equation sin(x+y) = sin(x) + sin(y) is not an identity because it does not match the correct trigonometric identity for the sine of a sum, which is sin(a + b) = sin(a)cos(b) + cos(a)sin(b). Using specific angles as an example, such as x and y both being 45°, clearly demonstrates that this equation does not hold as an identity.
Step-by-step explanation:
To show that the equation sin(x+y) = sin(x) + sin(y) is not an identity, we need to recall the correct formula for the sine of a sum, which is given by:
sin(a + b) = sin(a)cos(b) + cos(a)sin(b)
The equation given in the question sin(x+y) = sin(x) + sin(y) does not match this formula, which indicates it is not an identity. To further demonstrate this, we can use specific values for x and y. For example, let's take x = y = 45°:
sin(45° + 45°) = sin(90°) = 1
sin(45°) + sin(45°) = √2/2 + √2/2 = √2
Clearly, 1 does not equal √2, which means the equation is not an identity. This is also supported by trigonometric identities and triangles laws such as the Law of Sines and the Law of Cosines, which provide the proper ways to relate the angles and sides of a triangle, but do not support the incorrect equation provided.