What is the identity being proved?

[Silver Bolt]

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

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

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

$=\frac{\frac{\cos^2\left({A}\right)}{\sin\left({A}\right)}}{\frac{\cos\left({A}\right)+\cos\left({A}\right)\sin\left({A}\right)}{\sin\left({A}\right)}}$

$=\frac{\cos^2\left({A}\right)}{\cos\left({A}\right)+\cos\left({A}\right)\sin\left({A}\right)}$

What should be done from here?

 

[kaliprasad]

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

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

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

$=\frac{\frac{\cos^2\left({A}\right)}{\sin\left({A}\right)}}{\frac{\cos\left({A}\right)+\cos\left({A}\right)\sin\left({A}\right)}{\sin\left({A}\right)}}$

$=\frac{\cos^2\left({A}\right)}{\cos\left({A}\right)+\cos\left({A}\right)\sin\left({A}\right)}$

What should be done from here?

to continue

$=\frac{\cos^2\left({A}\right)}{\cos\left({A}\right)( 1+\sin\left({A}\right))}$
$= \frac{\cos\left({A}\right)}{( 1+\sin\left({A}\right))}$
$= \frac{\cos\left({A}\right)(( 1-\sin\left({A}\right))}{( 1+\sin\left({A}\right))( 1-\sin\left({A}\right))}$ multiply by 1- sin for the case of 1 + sin so on
$= \frac{\cos\left({A}\right)(( 1-\sin\left({A}\right))}{( 1- \sin^2\left({A}\right))}$
$= \frac{\cos\left({A}\right)(( 1-\sin\left({A}\right))}{( \cos^2\left({A}\right))}$
$= \frac{(( 1-\sin\left({A}\right))}{ \cos\left({A}\right)}$
$= \frac{\cot(A) (( 1-\sin\left({A}\right))}{\cot\left({A}\right) \cos\left({A}\right)}$
Now for the numerator
$= \cot(A) (( 1-\sin\left({A}\right))$
$= \cot(A) - \cot(A)\sin\left({A}\right)$
$= \cot(A) - \cos(A)$

hence the result

 

[Greg]

$$\frac{\cot(A)\cos(A)}{\cot(A)+\cos(A)}\cdot\frac{\tan(A)}{\tan(A)}=\frac{\cos(A)}{1+\sin(A)}$$

$$=\frac{\cos(A)(1-\sin(A))}{\cos^2(A)}=\frac{1-\sin(A)}{\cos(A)}\cdot\frac{\cot(A)}{\cot(A)}=\frac{\cot(A)-\cos(A)}{\cot(A)\cos(A)}$$