First slide

Multiple and sub- multiple Angles

Question

If $\cos θ = \frac{1}{2} (x + \frac{1}{x})$ then $\frac{1}{2} (x^{2} + \frac{1}{x^{2}})$ is equal to

Moderate

A

sin 2 $θ$
B

cos 2 $θ$
C

tan 2 $θ$
D

None of these

Solution

Given that, $\cos θ = \frac{1}{2} (x + \frac{1}{x})$
$\Rightarrow x + \frac{1}{x} = 2 \cos θ - - - - (i)$
We know that,
$\begin{array}{l} x^{2} + \frac{1}{x^{2}} = {(x + \frac{1}{x})}^{2} - 2 \\ = (2 \cos θ)^{2} - 2 = 4 \cos^{2} θ - 2 \\ = 2 \cos 2 θ \\ ∴ \frac{1}{2} (x^{2} + \frac{1}{x^{2}}) = \frac{1}{2} \times 2 \cos 2 θ = \cos 2 θ \end{array}$

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