173
If $\sin A + \cos A = \sqrt{3}$, then prove that $\tan A + \cot A = 1$.
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Given $\sin A + \cos A = \sqrt{3}$
Squaring both sides
$\sin^2A + \cos^2A + 2 \sin A \cos A = 3$
$\Rightarrow \sin A \cos A = 1$
$\frac{1}{\sin A \cos A} = 1$
$\frac{\sin^2A+\cos^2A}{\sin A \cos A} = 1$
$\therefore \tan A + \cot A = 1$
Squaring both sides
$\sin^2A + \cos^2A + 2 \sin A \cos A = 3$
$\Rightarrow \sin A \cos A = 1$
$\frac{1}{\sin A \cos A} = 1$
$\frac{\sin^2A+\cos^2A}{\sin A \cos A} = 1$
$\therefore \tan A + \cot A = 1$