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]
[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]
