Question

Verify that the equation is an identity. (Hint sin 2 x=sin (x+x) ) sin 2 x=2 sin x cos x Substitute 2 x=x+x and apply the sine of a sum identity.

Answer 1

To verify the equation sin 2x = sin(x+x) is an identity, we can substitute 2x = x+x and apply the sine of a sum identity. After applying the identity, we can conclude that sin 2x = 2 sin(x) cos(x) is an identity.

Step-by-step explanation:

To verify the equation sin 2x = sin(x+x) is an identity, we can substitute 2x = x+x and apply the sine of a sum identity. Let's start by substituting x+x for 2x in the equation:

sin(x+x) = 2 sin(x) cos(x)

Now, let's apply the sine of a sum identity:

sin(x+x) = sin(x)cos(x) + cos(x)sin(x)

Since both sides of the equation are equal, we can conclude that sin 2x = 2 sin(x) cos(x) is an identity.

Answer 2

The given equation is verified as an identity sin(2x) = 2sin(x)cos(x)

How to prove identity of an expression.

The sine of a sum identity, also known as the angle sum formula for sine, is given by:

sin(a + b) = sin(a)cos(b) + cos(a)sin(b)

This formula expresses the sine of the sum of two angles (a + b) in terms of the sines and cosines of the individual angles (a) and (b)

Verifying the identity sin(2x) = 2sin(x)cos(x) using the given hint, substitute 2x = x + x and apply the sine of a sum identity:

sin(2x) = sin(x + x)

Now, use the sine of a sum identity

sin(a + b) = sin(a)cos(b) + cos(a)sin(b)

sin(x + x) = sin(x)cos(x) + cos(x)sin(x)

Combine like terms:

sin(2x) = 2sin(x)cos(x)

Thus, the given equation is verified as an identity.