If \(\cot A = \sqrt{3}\), what is the value of \((1+\sin A)(1+\cos A)\)?
\(\dfrac{6+3\sqrt{3}}{4}\)
Step 1 — identify angle A.
\(\cot A = \sqrt{3} \;\Longrightarrow\; \tan A = \dfrac{1}{\sqrt{3}} \;\Longrightarrow\; A = 30^\circ\)
Step 2 — standard values:
\(\sin 30^\circ = \dfrac{1}{2},\quad \cos 30^\circ = \dfrac{\sqrt{3}}{2}\)
Step 3 — compute the product:
\((1+\sin A)(1+\cos A) = \left(1+\dfrac{1}{2}\right)\!\left(1+\dfrac{\sqrt{3}}{2}\right)\)
\(= \dfrac{3}{2} \cdot \dfrac{2+\sqrt{3}}{2} = \dfrac{3(2+\sqrt{3})}{4} = \dfrac{6+3\sqrt{3}}{4}\)
Hence the value is \(\dfrac{6+3\sqrt{3}}{4}\) — option (2).
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