Which expression is equal to [tex]$2 \sin \left(\frac{\pi}{10}\right) \cos \left(\frac{\pi}{10}\right)$[/tex]?

A. [tex][tex]$-\sin \left(\frac{\pi}{20}\right)$[/tex][/tex]
B. [tex]$-\sin \left(\frac{\pi}{5}\right)$[/tex]
C. [tex]$\sin \left(\frac{\pi}{5}\right)$[/tex]
D. [tex][tex]$\sin \left(\frac{\pi}{20}\right)$[/tex][/tex]



Answer :

To solve the problem of finding the expression equal to [tex]\(2 \sin \left(\frac{\pi}{10}\right) \cos \left(\frac{\pi}{10}\right)\)[/tex], we can use a trigonometric identity. Specifically, we use the double-angle identity for sine:

[tex]\[ 2 \sin(a) \cos(a) = \sin(2a) \][/tex]

Here, the variable [tex]\(a\)[/tex] is [tex]\(\frac{\pi}{10}\)[/tex]. Applying this identity to our expression:

[tex]\[ 2 \sin \left(\frac{\pi}{10}\right) \cos \left(\frac{\pi}{10}\right) = \sin \left(2 \times \frac{\pi}{10}\right) \][/tex]

Simplify the argument of the sine function:

[tex]\[ 2 \times \frac{\pi}{10} = \frac{2\pi}{10} = \frac{\pi}{5} \][/tex]

Therefore:

[tex]\[ 2 \sin \left(\frac{\pi}{10}\right) \cos \left(\frac{\pi}{10}\right) = \sin \left(\frac{\pi}{5}\right) \][/tex]

Among the given choices:
[tex]\[ -\sin \left(\frac{\pi}{20}\right) \][/tex]
[tex]\[ -\sin \left(\frac{\pi}{5}\right) \][/tex]
[tex]\[ \sin \left(\frac{\pi}{5}\right) \][/tex]
[tex]\[ \sin \left(\frac{\pi}{20}\right) \][/tex]

The expression that matches our result [tex]\(\sin \left(\frac{\pi}{5}\right)\)[/tex] is:

[tex]\[ \boxed{\sin \left(\frac{\pi}{5}\right)} \][/tex]