Question

Show that the equation is not an identity. sin(2x)=2sin(x)

Answer 1

The equation sin(2x) = 2sin(x) is not an identity.

Step-by-step explanation:

To show that the equation sin(2x) is not an identity, we can use the double angle identity for sine which states that sin(2x) = 2sin(x)cos(x). By comparing this to the given equation, we can see that they are not equal. Therefore, the equation sin(2x) = 2sin(x) is not an identity.

Answer 2

The equation sin(2x) = 2sin(x) is not an identity, as it fails for certain values, such as x = π/2.

Step-by-step explanation:

To verify whether the given equation is an identity, we need to check if it holds true for all values of x. If the equation is true for some values and false for others, it is not an identity. Let's start by simplifying both sides of the equation:

Using the double-angle identity for sine, sin(2x) can be expressed as 2sin(x)cos(x).

Now, we compare 2sin(x)cos(x) with 2sin(x). If this equation were an identity, both sides would be equivalent for all x values. However, it becomes clear that for this equation to be true, cos(x) would need to be 1 for all x, which is not the case.

Consider x = π/2. On the left side, sin(2x) becomes sin(π) = 0, while on the right side, 2sin(x) becomes 2. Since 0 ≠ 2, the equation is false for x = π/2, proving that it is not an identity.

In summary, the equation sin(2x) = 2sin(x) is not an identity because it fails to hold true for all values of x. It's crucial to find counterexamples to demonstrate when the equation does not satisfy the equality. This process ensures a thorough understanding of the limitations of the given expression.