Verify a trigonometric relation

Question

Suppose we have the ratio $\frac{a}{\sqrt{a-b}}$, and we have that $b=a\cos(c)$. Then, do we have $\frac{a}{\sqrt{a-b}}=\csc(\frac{c}{2})$? Or at least some some thing with cosecant?

Answer 1

$$\frac{a}{\sqrt{a-b}}=\frac{a}{\sqrt{a-a\cos c}}= \frac{a}{\sqrt{a} \times \sqrt{1-\cos c}}$$

Now use $$1-\cos c=2 \sin^2 \left( \frac c2 \right)$$

To get $$ \frac{a}{\sqrt{a} \times \sqrt{1-\cos c}}= \frac{\sqrt a}{\sqrt{2 \sin^2 \left( \frac c2 \right)}}=\color{blue}{\sqrt{\frac a2} \Bigg|\csc \left( \frac c2 \right) \Bigg|}$$