We all know that from Euler's reflection formula $$\Gamma(m) \Gamma(1-m) = \frac{\pi}{\sin(m \pi)}.$$
But while solving problems there are circumstances where one often ends with a quotient of the form $$\frac{\Gamma(m)}{\Gamma (1-m)}.$$ What will be the simplification of it? As an example, how can one simplify $\dfrac{\Gamma(1/4)}{\Gamma(3/4)}$?
Unfortunately, in general there is no nice simplification formula for a quotient between two Gamma functions of the form $\Gamma (m)/\Gamma (1 - m)$. The best one can do is to use Euler's reflection formula of $$\Gamma (m) \Gamma (1 - m) = \frac{\pi}{\sin (m \pi)},$$ and write the quotient $\Gamma (m)/\Gamma (1 - m)$ in terms of a single Gamma function as follows: $$\frac{\Gamma (m)}{\Gamma (1 - m)} = \frac{\Gamma^2 (m) \sin (m \pi)}{\pi}.$$
In the case of the example you give, we have $$\frac{\Gamma \left (\frac{1}{4} \right )}{\Gamma \left (\frac{3}{4} \right )} = \frac{\Gamma \left (\frac{1}{4} \right )}{\Gamma \left (1 - \frac{1}{4} \right )} = \frac{1}{\pi} \Gamma^2 \left (\frac{1}{4} \right ) \sin \left (\frac{\pi}{4} \right ) = \frac{1}{\pi \sqrt{2}} \Gamma^2 \left (\frac{1}{4} \right ).$$
