Question

cos2pi/15cos4pi/15cos8pi/15cos16pi/15= 1/16​

Answer 1

Answer:

$\displaystyle{\cos\dfrac{2\pi}{15}\cos\dfrac{4\pi}{15}\cos\dfrac{8\pi}{15}\cos\dfrac{16\pi}{15}=\dfrac{1}{16} \ [Proved]}$

Step-by-step explanation:

$\displaystyle{\cos\dfrac{2\pi}{15}\cos\dfrac{4\pi}{15}\cos\dfrac{8\pi}{15}\cos\dfrac{16\pi}{15}=\dfrac{1}{16}}\\\\\\\displaystyle{Rewrite \ as}\\\\\displaystyle{\cos\dfrac{2\pi}{15}\cos\dfrac{2^2\pi}{15}\cos\dfrac{2^3\pi}{15}\cos\dfrac{2^4\pi}{15}=\dfrac{1}{16}}\\\\\\\displaystyle{We \ have \ formula}\\\\\\\displaystyle{\cos 2A\cos2^2A\cos2^3A.. \ ..\cos^{n-1}A=\dfrac{\sin2^nA}{2^n\sin A}}$

$\displaystyle{\cos\dfrac{2\pi}{15}\cos\dfrac{2^2\pi}{15}\cos\dfrac{2^3\pi}{15}\cos\dfrac{2^4\pi}{15}=\dfrac{1}{16}}\\\\\\\displaystyle{L.H.S=\cos\dfrac{2\pi}{15}\cos\dfrac{2^2\pi}{15}\cos\dfrac{2^3\pi}{15}\cos\dfrac{2^4\pi}{15}}\\\\\\\displaystyle{Applying \ formula \ where \ A=\dfrac{\pi}{15}}\\\\\\\displaystyle{L.H.S=\dfrac{\sin2^4\dfrac{\pi}{15}}{2^4\sin\dfrac{\pi}{15}}}\\\\\\\displaystyle{L.H.S=\dfrac{1}{16}\left(\frac{\sin192}{\sin12}\right)}$

$\displaystyle{L.H.S=\dfrac{1}{16}\left(\frac{\sin(180+12)}{\sin12}\right)}\\\\\\\displaystyle{L.H.S=\dfrac{1}{16}\left(\frac{\sin12}{\sin12}\right)}\\\\\\\displaystyle{L.H.S=\dfrac{1}{16}}$

L.H.S. = R.H.S.

Hence proved .