First slide

Multiple and sub- multiple Angles

Question

If $\tan x = b / a, then \sqrt{(a + b) / (a - b)} + \sqrt{(a - b) / (a + b)}$ is equal to $(where x \in (0, \frac{π}{2}), a > b > 0)$

Moderate

A

$2 \sin x / \sqrt{\sin 2 x}$
B

$2 \cos x / \sqrt{\cos 2 x}$
C

$2 \cos x / \sqrt{\sin 2 x}$
D

$2 \sin x / \sqrt{\cos 2 x}$

Solution

we have $\sqrt{(\frac{a + b}{a - b})} + \sqrt{(\frac{a - b}{a + b})} = \frac{a + b + a - b}{\sqrt{(a^{2} - b^{2})}}$
$= \frac{2 a}{\sqrt{(a^{2} - b^{2})}} = \frac{2}{\sqrt{[1 - (b / a)^{2}]}}$
$\begin{matrix} = \frac{2}{\sqrt{(1 - \tan^{2} x)}} = \frac{2 \cos x}{\sqrt{(\cos^{2} x - \sin^{2} x)}} \\ = \frac{2 \cos x}{\sqrt{(\cos 2 x)}} \end{matrix}$

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