Prove the following trigonometric identities.

(i) We have to prove $\frac{\cot A+\tan B}{\cot B+\tan A}=\cot A \tan B$

Now,

$\frac{\cot A+\tan B}{\cot B+\tan A}=\frac{\cot A+\frac{1}{\cot B}}{\cot B+\frac{1}{\cot A}}$

$=\frac{\frac{\cot A \cot B+1}{\cot B}}{\frac{\cot A \cot B+1}{\cot A}}$

$=\frac{\cot A}{\cot B}$

$=\cot A \frac{1}{\cot B}$

$=\cot A \tan B$

Hence proved.

(ii) We have to prove $\frac{\tan A+\tan B}{\cot A+\cot B}=\tan A \tan B$

Now,

$\frac{\tan A+\tan B}{\cot A+\cot B}=\frac{\tan A+\tan B}{\frac{1}{\tan A}+\frac{1}{\tan B}}$

$=\frac{\tan A+\tan B}{\frac{\tan B+\tan A}{\tan A \tan B}}$

$=\tan A \tan B$

Hence proved.