Prove Identity: $\frac{\cos A - \sin A}{\cos A + \sin A}$

[Silver Bolt]

$\frac{1-\tan\left({A}\right)}{1+\tan\left({A}\right)}=\frac{\cot\left({A}\right)-1}{\cot\left({A}\right)+1}$

$L..H.S=\frac{1-\frac{\sin\left({A}\right)}{\cos\left({A}\right)}}{1+\frac{\sin\left({A}\right)}{\cos\left({A}\right)}}$

$=\frac{\cos\left({A}\right)-\sin\left({A}\right)}{\cos\left({A}\right)+\sin\left({A}\right)}$

What should be done from here?

 

[I like Serena]

$\frac{1-\tan\left({A}\right)}{1+\tan\left({A}\right)}=\frac{\cot\left({A}\right)-1}{\cot\left({A}\right)+1}$

$L..H.S=\frac{1-\frac{\sin\left({A}\right)}{\cos\left({A}\right)}}{1+\frac{\sin\left({A}\right)}{\cos\left({A}\right)}}$

$=\frac{\cos\left({A}\right)-\sin\left({A}\right)}{\cos\left({A}\right)+\sin\left({A}\right)}$

What should be done from here?

Hi Silver Bolt! ;)

Can we divide both the numerator and the denominator by $\sin A$?

 

[Greg]

$$\frac{1-\tan(A)}{1+\tan(A)}\cdot\frac{\cot(A)}{\cot(A)}$$