Prove that $\lim_{a \to 0^{+}} \int_{0}^{a} \frac{1}{\sqrt{\cos(x)-\cos(a)}} \;dx=\frac{\pi}{\sqrt{2}}$

You can expand $\cos(x)$ into a Taylor series, keeping only the first two terms, since all the rest go to zero. You get the integral: $$\int_0^b{\frac{\sqrt{2}}{\sqrt{a^2 - x^2}}}dx = \sqrt{2} \arctan\left(\frac{b}{\sqrt{a^2-b^2}}\right)$$ Since $\lim_{x\rightarrow \infty}{\arctan {x}} = \pi/2$, the result follows.

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

$$ \sqrt {2}\int _{0}^{1}\!{\frac {1}{\sqrt {1-{y}^{2}}}}{dx} = \frac{\pi}{\sqrt{2}} \,.$$