TITLE: Prove that $\lim_{a \to 0^{+}} \int_{0}^{a} \frac{1}{\sqrt{\cos(x)-\cos(a)}} \;dx=\frac{\pi}{\sqrt{2}}$
QUESTION [15 upvotes]: How can I prove that
$$\lim_{a \to 0^{+}} \int_{0}^{a} \frac{1}{\sqrt{\cos(x)-\cos(a)}} \;dx=\frac{\pi}{\sqrt{2}}$$

REPLY [8 votes]: Exploiting the trigonometric identity $ \cos(x) = 1-2\sin^2(\frac{x}{2}) $ and making the change of variables $x=ay$ results in the following integral,
$$ a\int _{0}^{1}\!{\frac {1}{\sqrt {-2\, \sin^2 \left( \frac{a}{2}  \right)  + 2\, \sin^2 \left( \frac{a y}{2}  
 \right) }}}{dy}$$
Using the fact that $ \sin(a)\approx a $ when $a \rightarrow 0 $ in the above integral yields
$$ \sqrt {2}\int _{0}^{1}\!{\frac {1}{\sqrt {1-{y}^{2}}}}{dx} = \frac{\pi}{\sqrt{2}} \,.$$