Question

Select the correct answer.

Rewrite 2 cos 75° sin 75° using a double-angle identity.
A. 2 sin 75°
B. cos 75°
C. sin 150°
D. cos 150°
E. sin 75°

Answer 1

The expression
`\(2 \cos 75^\circ \sin 75^\circ\)` can be rewritten using the double-angle identity as
`\(\sin 150^\circ\)`. Therefore, the answer is C.

To rewrite
`\(2 \cos 75^\circ \sin 75^\circ\)` using a double-angle identity, we can use the formula
`\( \sin 2\theta = 2\sin \theta \cos \theta\)`. In this case,
`\(\theta = 75^\circ\)`. So,

`\[ 2 \cos 75^\circ \sin 75^\circ = 2 \cdot \sin 2 \cdot (75^\circ) \]`

Now, the double-angle identity for sine is
`\( \sin 2\theta = 2\sin \theta \cos \theta \)`. Therefore,

`\[ 2 \cdot \sin 2 \cdot (75^\circ) = \sin (2 \cdot 75^\circ) \]`

Now, simplify the expression inside the sine function:

`\[ \sin (2 \cdot 75^\circ) = \sin 150^\circ \]`

Therefore, the given expression
`\(2 \cos 75^\circ \sin 75^\circ\)` can be rewritten as

`\(\sin 150^\circ\)`. Therefore, the correct answer is:

`\[ \text{C.} \ \sin 150^\circ \]`