The required "option A) [tex]\dfrac{1}{2} (\cos B-\sqrt{3}\sin B)[/tex]" is correct.
Step-by-step explanation:
We have,
[tex]\cos(\dfrac{\pi}{3}+B)[/tex]
To find, the value of [tex]\cos(\dfrac{\pi}{3}+B)[/tex] = ?
∴ [tex]\cos(\dfrac{\pi}{3}+B)[/tex]
= [tex]\cos(60+B)[/tex]
We know that,
[tex]\cos(A+B)=\cos Acos B-\sin A\sin B[/tex]
∴ [tex]\cos(60+B)[/tex]
[tex]=\cos 60\cos B-\sin 60\sin B[/tex]
= [tex]\dfrac{1}{2} \cos B-\dfrac{\sqrt{3}}{2}\sin B[/tex]
Taking [tex]\dfrac{1}{2}[/tex] as common, we get
= [tex]\dfrac{1}{2} (\cos B-\sqrt{3}\sin B)[/tex]
Thus, the required "option A) [tex]\dfrac{1}{2} (\cos B-\sqrt{3}\sin B)[/tex]" is correct.
