If \(\tan A + \cot A = 2\) and \(A \in (0,\ \pi/2)\), then what is \(\sin A + \cos A\)?
\(\sqrt{2}\)
Step 1 – solve \(\tan A + \cot A = 2\):
\(\tan A + \dfrac{1}{\tan A} = 2\)
Multiply both sides by \(\tan A\):
\(\tan^2 A - 2\tan A + 1 = 0 \implies (\tan A - 1)^2 = 0 \implies \tan A = 1\).
Step 2 – find A:
In \((0,\ \pi/2)\), \(\tan A = 1 \implies A = \dfrac{\pi}{4} = 45^{\circ}\).
Step 3 – evaluate:
\(\sin 45^{\circ} + \cos 45^{\circ} = \dfrac{1}{\sqrt{2}} + \dfrac{1}{\sqrt{2}} = \dfrac{2}{\sqrt{2}} = \sqrt{2}\).
Hence the answer is \(\sqrt{2}\).
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