Prove that: $\frac{\sqrt{1-\cos A}}{\sqrt{1+\cos A}}$  = 𝑐𝑜𝑠𝑒𝑐 𝐴 − 𝑐𝑜𝑡𝐴.

We have to prove whether LHS is equal to RHS or not.
𝐿𝐻𝑆 =$\frac{\sqrt{1-\cos A}}{\sqrt{1+\cos A}}$

Multiplying LHS by $\frac{\sqrt{1-\cos A}}{\sqrt{1-\cos A}}$
 

⇒ 𝐿𝐻𝑆 = $\frac{\sqrt{1-\cos A}}{\sqrt{1+\cos A}}$ × $\frac{\sqrt{1-\cos A}}{\sqrt{1-\cos A}}$
 

⇒ 𝐿𝐻S = $\frac{\sqrt{{(1-\cos A)}^2}}{\sqrt{1+\cos A\times 1-\cos A}}$
⇒ 𝐿𝐻𝑆 = $\frac{1-\cos A}{\sqrt{1-{\cos }^{2}A}}$
As 𝑠𝑖𝑛²𝐴 + 𝑐𝑜𝑠²𝐴 = 1
⇒ 1 − 𝑐𝑜𝑠²𝐴 = 𝑠𝑖𝑛²𝐴
Putting this in LHS, we get
⇒ 𝐿𝐻𝑆 = $\frac{1-\cos A}{\sin A}$

⇒ 𝐿𝐻𝑆 =    $\frac{1}{\sin A}-\frac{\cos A}{\sin A}$
⇒ 𝐿𝐻𝑆 = 𝑐𝑜𝑠𝑒𝑐 𝐴 − 𝑐𝑜𝑡 𝐴
⇒ 𝐿𝐻𝑆 = 𝑅𝐻𝑆
Hence, proved.