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Question: The value of \[\sin \dfrac{\pi }{14}\cdot \sin \dfrac{3\pi }{14}\cdot \sin \dfrac{5\pi }{14}\cdot \s...

The value of sin⁡π14⋅sin⁡3π14⋅sin⁡5π14⋅sin⁡7π14⋅sin⁡9π14⋅sin⁡11π14⋅sin⁡13π14\sin \dfrac{\pi }{14}\cdot \sin \dfrac{3\pi }{14}\cdot \sin \dfrac{5\pi }{14}\cdot \sin \dfrac{7\pi }{14}\cdot \sin \dfrac{9\pi }{14}\cdot \sin \dfrac{11\pi }{14}\cdot \sin \dfrac{13\pi }{14} is equal to:
a. 1
b. 116\dfrac{1}{16}
c. 164\dfrac{1}{64}
d. None of These

Explanation

Solution

Hint: Express the obtuse angles in forms of smaller acute angles of sine. Then reduce it to simplified forms by changing sine to cosine.

Complete step-by-step answer:

Let, S = sin⁡π14⋅sin⁡3π14⋅sin⁡5π14⋅sin⁡7π14⋅sin⁡9π14⋅sin⁡11π14⋅sin⁡13π14\sin \dfrac{\pi }{14}\cdot \sin \dfrac{3\pi }{14}\cdot \sin \dfrac{5\pi }{14}\cdot \sin \dfrac{7\pi }{14}\cdot \sin \dfrac{9\pi }{14}\cdot \sin \dfrac{11\pi }{14}\cdot \sin \dfrac{13\pi }{14}
Now we know that sin⁡x=sin⁡(π−x)\sin x=\sin \left( \pi -x \right) for all real values of x.

So, S = sin⁡π14⋅sin⁡3π14⋅sin⁡5π14⋅sin⁡7π14⋅sin⁡(π−9π14)⋅sin⁡(π−11π14)⋅sin⁡(π−13π14)\sin \dfrac{\pi }{14}\cdot \sin \dfrac{3\pi }{14}\cdot \sin \dfrac{5\pi }{14}\cdot \sin \dfrac{7\pi }{14}\cdot \sin \left( \pi -\dfrac{9\pi }{14} \right)\cdot \sin \left( \pi -\dfrac{11\pi }{14} \right)\cdot \sin \left( \pi -\dfrac{13\pi }{14} \right)
⇒\Rightarrow S = sin⁡π14⋅sin⁡3π14⋅sin⁡5π14⋅sin⁡7π14⋅sin⁡5π14⋅sin⁡3π14⋅sin⁡π14\sin \dfrac{\pi }{14}\cdot \sin \dfrac{3\pi }{14}\cdot \sin \dfrac{5\pi }{14}\cdot \sin \dfrac{7\pi }{14}\cdot \sin \dfrac{5\pi }{14}\cdot \sin \dfrac{3\pi }{14}\cdot \sin \dfrac{\pi }{14}

Now we know that sin⁡7π14=sin⁡π2=1\sin \dfrac{7\pi }{14}=\sin \dfrac{\pi }{2}=1.
⇒\Rightarrow S = (sin⁡π14⋅sin⁡3π14⋅sin⁡5π14)2{{\left( \sin \dfrac{\pi }{14}\cdot \sin \dfrac{3\pi }{14}\cdot \sin \dfrac{5\pi }{14} \right)}^{2}}

We know that sin⁡x=cos⁡(π2−x)\sin x=\cos \left( \dfrac{\pi }{2}-x \right) for all real values of x.
So, now S = (sin⁡π14⋅cos⁡(π2−3π14)⋅cos⁡(π2−5π14))2{{\left( \sin \dfrac{\pi }{14}\cdot \cos \left( \dfrac{\pi }{2}-\dfrac{3\pi }{14} \right)\cdot \cos \left( \dfrac{\pi }{2}-\dfrac{5\pi }{14} \right) \right)}^{2}}
⇒\Rightarrow S = (sin⁡π14⋅cos⁡2π7⋅cos⁡π7)2{{\left( \sin \dfrac{\pi }{14}\cdot \cos \dfrac{2\pi }{7}\cdot \cos \dfrac{\pi }{7} \right)}^{2}}

We know from multiple angle formulas that 2sin⁡xcos⁡x=sin⁡2x2\sin x\cos x=\sin 2x for all real values of x.
⇒\Rightarrow S = (12cos⁡π14⋅2sin⁡π14⋅cos⁡π14⋅cos⁡2π7⋅cos⁡π7)2{{\left( \dfrac{1}{2\cos \dfrac{\pi }{14}}\cdot 2\sin \dfrac{\pi }{14}\cdot \cos \dfrac{\pi }{14}\cdot \cos \dfrac{2\pi }{7}\cdot \cos \dfrac{\pi }{7} \right)}^{2}} [Multiplying numerator & denominator by same]
⇒\Rightarrow S = (12cos⁡π14⋅sin⁡π7⋅cos⁡2π7⋅cos⁡π7)2{{\left( \dfrac{1}{2\cos \dfrac{\pi }{14}}\cdot \sin \dfrac{\pi }{7}\cdot \cos \dfrac{2\pi }{7}\cdot \cos \dfrac{\pi }{7} \right)}^{2}} [Applying the above double angle formula]
⇒\Rightarrow S = (14cos⁡π14⋅2sin⁡π7⋅cos⁡π7⋅cos⁡2π7)2{{\left( \dfrac{1}{4\cos \dfrac{\pi }{14}}\cdot 2\sin \dfrac{\pi }{7}\cdot \cos \dfrac{\pi }{7}\cdot \cos \dfrac{2\pi }{7} \right)}^{2}} [Repeating the process]
⇒\Rightarrow S = (14cos⁡π14⋅sin⁡2π7⋅cos⁡2π7)2{{\left( \dfrac{1}{4\cos \dfrac{\pi }{14}}\cdot \sin \dfrac{2\pi }{7}\cdot \cos \dfrac{2\pi }{7} \right)}^{2}} [Applying the above double angle formula]
⇒\Rightarrow S = (18cos⁡π14⋅2sin⁡2π7⋅cos⁡2π7)2{{\left( \dfrac{1}{8\cos \dfrac{\pi }{14}}\cdot 2\sin \dfrac{2\pi }{7}\cdot \cos \dfrac{2\pi }{7} \right)}^{2}} [Repeating the process]
⇒\Rightarrow S = (18cos⁡π14⋅sin⁡4π7)2{{\left( \dfrac{1}{8\cos \dfrac{\pi }{14}}\cdot \sin \dfrac{4\pi }{7} \right)}^{2}} [Applying the above double angle formula]

Now sin⁡4π7=cos⁡(π2−4π7)=cos⁡(−π14)\sin \dfrac{4\pi }{7}=\cos \left( \dfrac{\pi }{2}-\dfrac{4\pi }{7} \right)=\cos \left( -\dfrac{\pi }{14} \right).

However, cosine is an even function. So cos⁡(−x)=cos⁡x\cos (-x)=\cos x for all real values of x.
⇒\Rightarrow S = (18cos⁡π14⋅cos⁡π14)2{{\left( \dfrac{1}{8\cos \dfrac{\pi }{14}}\cdot \cos \dfrac{\pi }{14} \right)}^{2}} = (18)2=164{{\left( \dfrac{1}{8} \right)}^{2}}=\dfrac{1}{64}.

Hence, we get the value of S. This is the required value for the given product.

Hence, the correct answer to the given question is option (c) 164\dfrac{1}{64}.

Note: Expressing large multiples into smaller ones will lead the process simpler and make the calculations easier. You can also assume the common value as x and apply the same rules of multiples angles, and finally put back the value to get the desired result.