question_answer

. In \[\Delta ABC\] right-angled at point B, if \[\tan A=\frac{1}{\sqrt{3}},\]then the value of \[\sin A.\cos C+\cos A.\sin C\]is:   (NCERT EXEMPLAR)

A) 0

B) 1

C) -1

D) \[\frac{1}{2}\]

Correct Answer: B

Solution :

[b] \[\tan A=\frac{1}{\sqrt{3}}=\tan 30{}^\circ \]

\[\Rightarrow \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,A=30{}^\circ \]

\[\therefore \,\,\,\,\,\,\,\sin A\cdot \cos C+\cos A.\sin C\]

\[=\sin 30{}^\circ \cdot \cos 60{}^\circ +\cos 30{}^\circ \cdot \sin 60{}^\circ \]

\[=\frac{1}{2}\times \frac{1}{2}+\frac{\sqrt{3}}{2}\times \frac{\sqrt{3}}{2}\]

\[=\frac{1}{4}+\frac{3}{4}=\frac{4}{4}=1\]