Question

If $(2^n+1)\theta =\pi ,\text{then}{2}^{n}cos\theta cos2\theta \cdots cos{2}^2\theta \cdots 2^{n-1}\theta =$

• A. -1
• B. 1
• C. $\frac{1}{2}$
• D. none of these

Answer 1

The correct option is B

1

$(2^n+1)\theta =\pi$ (Given)

$\Rightarrow 2^n\theta +\theta =\pi \Rightarrow 2^n\theta =\pi -\theta \Rightarrow sin{2}^n\theta =sin(\pi -\theta )$

$\Rightarrow sin{2}^n\theta =sin\theta$ . . . (1)

$2^{n}cos\theta cos2\theta cos{2}^2\theta \cdots cos{2}^{n-1}\theta =2^n\times \frac{sin{2}^n\theta }{2^{n}sin\theta }=\frac{sin{2}^n\theta }{sin\theta }$
$=\frac{sin\theta }{sin\theta }$ [From (1)]

= 1