<s>[INST] Another trigonometric proof...? ...sigh..another problem how shall I prove the following?
$$ {\cot A\over1- \tan A} + {\tan A \over 1- \cot A} = 1 + \tan A + \cot A$$
so what now? the following's what I've done:
$$\cot A - \cot^2 A + \tan A- \tan^2 A \over 2 - \tan A - \cot A$$
 [/INST] Let $\tan A=a\implies \cot A=\frac1a$
So, the problem reduces to 
$$\frac{\frac1a}{1-a}+\frac a{1-\frac1a}=\frac1{a(1-a)}+\frac{a^2}{a-1}$$
$$=\frac1{a(1-a)}-\frac{a^2}{1-a}=\frac{1-a^3}{a(1-a)}=\frac{1+a+a^2}a=a+\frac1a+1$$

Alternatively,
$$\frac{\tan A}{1-\cot A}=\frac{\tan^2A}{\tan A-1} (\text{ multiplying the numerator & the denominator by }\tan A) $$
$$\implies \frac{\tan A}{1-\cot A}=-\frac{\tan^2A}{1-\tan A}$$
$$\text{So,} {\cot A\over1- \tan A} + {\tan A \over 1- \cot A}={\cot A\over1- \tan A} -\frac{\tan^2A}{1-\tan A}=\frac{\cot A-\tan^2A}{1-\tan A}=\frac{1-\tan^3A}{\tan A(1-\tan A)} (\text{ multiplying the numerator & the denominator by }\tan A) $$
$$=\frac{1+\tan A+\tan^2A}{\tan A}\text{ (assuming  }1-\tan A\ne0)$$
$$=\cot A+1+\tan A$$
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