Question

verify that the equation is an identity. (hint: sin 2x = sin(x + x))

sin 2x = 2 sin x cos x

substitute 2x = x + x and apply the sine of a sum identity.

sin 2x = sin(x + x)
= \square

Explanation:

Step1: Recall sine sum identity

The sine of a sum identity is $\sin(A + B) = \sin A \cos B + \cos A \sin B$. Here, $A = x$ and $B = x$.

Step2: Apply the identity

Substitute $A = x$ and $B = x$ into the identity:
$\sin(x + x) = \sin x \cos x + \cos x \sin x$

Step3: Simplify the expression

Combine like terms: $\sin x \cos x + \cos x \sin x = 2\sin x \cos x$

Answer:

$\sin(x + x)=\sin x\cos x+\cos x\sin x = 2\sin x\cos x$ (So the box should be filled with $\sin x\cos x+\cos x\sin x$ which simplifies to $2\sin x\cos x$)