Department of Pre-University Education, KarnatakaPUC Karnataka Science Class 11

# If ( 2 N + 1 ) X = π , Then 2 N Cos X Cos 2 X Cos 2 2 X . . . Cos 2 N − 1 X = 1 - Mathematics

MCQ

If  $\left( 2^n + 1 \right) x = \pi,$ then $2^n \cos x \cos 2x \cos 2^2 x . . . \cos 2^{n - 1} x = 1$

#### Options

• -1

• 1

• 1/2

• None of these

#### Solution

1

$\left( 2^n + 1 \right)x = \pi \left( \text{ Given } \right)$

$\Rightarrow 2^n x + x = \pi$

$\Rightarrow 2^n x = \pi - x$

$\Rightarrow \sin 2^n x = \sin\left( \pi - x \right)$

$\Rightarrow \sin 2^n x = \sin x . . . (1)$

$2^n \cos x \cos 2x \cos 2^2 x . . . \cos 2^{n - 1} x = 2^n \times \frac{\sin 2^n x}{2^n \sin x}$
$= \frac{\sin 2^n x}{\sin x}$
$= \frac{\sin x}{\sin x} \left[ \text{ From } (1) \right]$

Concept: Values of Trigonometric Functions at Multiples and Submultiples of an Angle
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#### APPEARS IN

RD Sharma Class 11 Mathematics Textbook
Chapter 9 Values of Trigonometric function at multiples and submultiples of an angle
Q 27 | Page 45