If =1 2 (x+1 x ), then 1 2 (x^2+1 x^2 )= - Vidyadip

MCQ

If $\cos \theta=\frac{1}{2}\left(x+\frac{1}{x}\right)$, then $\frac{1}{2}\left(x^2+\frac{1}{x^2}\right)=$

A
$\sin 2 \theta$
✓
$\cos 2 \theta$
C
$\tan 2 \theta$
D
$\sec 2 \theta$

✓

Answer

Correct option: B.

$\cos 2 \theta$

(B)
Given that, $\cos \theta=\frac{1}{2}\left(x+\frac{1}{x}\right)$
$\Rightarrow x+\frac{1}{x}=2 \cos \theta$
Now, $x^2+\frac{1}{x^2}=\left(x+\frac{1}{x}\right)^2-2$
$=(2 \cos \theta)^2-2$
$=4 \cos ^2 \theta-2=2 \cos 2 \theta$
$\therefore \quad \frac{1}{2}\left(x^2+\frac{1}{x^2}\right)=\frac{1}{2} \times 2 \cos 2 \theta=\cos 2 \theta$

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