use the product - to - sum identities to rewrite the expression.\n2\\cos\\left(\\frac{\\pi}{5}\\right)\\sin\\…

use the product - to - sum identities to rewrite the expression.\n2\\cos\\left(\\frac{\\pi}{5}\\right)\\sin\\left(\\frac{\\pi}{9}\\right)\nwhich choice below is equivalent to 2\\cos\\left(\\frac{\\pi}{5}\\right)\\sin\\left(\\frac{\\pi}{9}\\right)?\n\\( \\bigcirc \\) a. \\( \\sin \\frac{14 \\pi}{45}-\\sin \\frac{4 \\pi}{45} \\)\n\\( \\bigcirc \\) b. \\( \\sin \\frac{14 \\pi}{45}+\\sin \\frac{4 \\pi}{45} \\)\n\\( \\bigcirc \\) c. \\( 2\\left(\\sin \\frac{14 \\pi}{45}-\\sin \\frac{4 \\pi}{45}\\right) \\)\n\\( \\bigcirc \\) d. \\( \\cos \\frac{14 \\pi}{45}-\\cos \\frac{4 \\pi}{45} \\)

use the product - to - sum identities to rewrite the expression.\n2\\cos\\left(\\frac{\\pi}{5}\\right)\\sin\\left(\\frac{\\pi}{9}\\right)\nwhich choice below is equivalent to 2\\cos\\left(\\frac{\\pi}{5}\\right)\\sin\\left(\\frac{\\pi}{9}\\right)?\n\\( \\bigcirc \\) a. \\( \\sin \\frac{14 \\pi}{45}-\\sin \\frac{4 \\pi}{45} \\)\n\\( \\bigcirc \\) b. \\( \\sin \\frac{14 \\pi}{45}+\\sin \\frac{4 \\pi}{45} \\)\n\\( \\bigcirc \\) c. \\( 2\\left(\\sin \\frac{14 \\pi}{45}-\\sin \\frac{4 \\pi}{45}\\right) \\)\n\\( \\bigcirc \\) d. \\( \\cos \\frac{14 \\pi}{45}-\\cos \\frac{4 \\pi}{45} \\)

Answer

Explanation:

Step1: Recall the product - to - sum identity

The identity (2\cos A\sin B=\sin(A + B)-\sin(A - B)).

Step2: Substitute (A=\frac{\pi}{5}) and (B = \frac{\pi}{9})

First, calculate (A + B=\frac{\pi}{5}+\frac{\pi}{9}=\frac{9\pi+5\pi}{45}=\frac{14\pi}{45}). Second, calculate (A - B=\frac{\pi}{5}-\frac{\pi}{9}=\frac{9\pi - 5\pi}{45}=\frac{4\pi}{45}). Then (2\cos\frac{\pi}{5}\sin\frac{\pi}{9}=\sin(\frac{\pi}{5}+\frac{\pi}{9})-\sin(\frac{\pi}{5}-\frac{\pi}{9})=\sin\frac{14\pi}{45}-\sin\frac{4\pi}{45}).

Answer:

A. (\sin\frac{14\pi}{45}-\sin\frac{4\pi}{45})