Question: Prove $\cot (A+ 45^{\circ}) = \dfrac{\csc(A)-\sec(A)}{\csc(A)+\sec(A)}$
My attempt:LHS
$$ \cot (A+ 45^{\circ})$$
$$ \frac{\cos(A+ 45^{\circ})}{\sin(A+45^{\circ})} $$
$$ \cos(45^{\circ}) = \frac{1}{\sqrt2},\sin(45^{\circ}) = \frac{1}{\sqrt2} $$
$$ \frac{\cos(A+ 45^{\circ})}{\sin(A+45^{\circ})} $$
$$ \frac{\cos(A)\cos(45^{\circ}) - \sin(A)\sin(45^{\circ})}{\sin(A)\cos(45^{\circ}) + \cos(A)\sin(45^{\circ})} $$
$$ \frac{ \frac{\cos(A)}{\sqrt2} - \frac{\sin(A)}{\sqrt2}} { \frac{\sin(A)}{\sqrt2} + \frac{\cos(A)}{\sqrt2}}$$
$$ \frac{\cos(A)-\sin(A)}{\sqrt2} \times \frac{\sqrt{2}}{\sin(A)+\cos(A)} $$
$$ \frac{\cos(A)-\sin(A)}{\sin(A)+\cos(A)} $$
Now I am stuck..