Respuesta :
Solution:
The question is given below as
[tex]\tan \mleft(\frac{\pi}{3}\mright)+\sin \mleft(\frac{5\pi}{6}\mright)\cos \mleft(-\frac{3\pi}{4}\mright)[/tex]Step 1:
Using the following identity below
[tex]\sin \mleft(x\mright)=\cos \mleft(\frac{\pi}{2}-x\mright)[/tex]By applying the identity above, we will have
[tex]\begin{gathered} \sin \mleft(\frac{5\pi}{6}\mright)=\cos \mleft(\frac{\pi}{2}-\frac{5\pi}{6}\mright) \\ \sin (\frac{5\pi}{6})=\cos (\frac{3\pi-5\pi}{6}) \\ \sin (\frac{5\pi}{6})=\cos (-\frac{2\pi}{6}) \\ \sin (\frac{5\pi}{6})=\cos (-\frac{\pi}{3}) \end{gathered}[/tex]Step 2:
Use the property below
[tex]\cos \mleft(-x\mright)=\cos \mleft(x\mright)[/tex]By applying the property above, we will have
[tex]\begin{gathered} \cos \mleft(-\frac{\pi}{3}\mright)=\cos \mleft(\frac{\pi}{3}\mright) \\ \cos \mleft(-\frac{3\pi}{4}\mright)=\cos \mleft(\frac{3\pi}{4}\mright) \end{gathered}[/tex]The question above then becomes
[tex]\begin{gathered} \tan (\frac{\pi}{3})+\sin (\frac{5\pi}{6})\cos (-\frac{3\pi}{4}) \\ \tan (\frac{\pi}{3})+\cos (\frac{\pi}{3})\text{.}\cos (\frac{3\pi}{4}) \end{gathered}[/tex]Using the following trivial identities, we will have
[tex]\begin{gathered} \tan \mleft(\frac{\pi}{3}\mright)=\sqrt{3} \\ \cos \mleft(\frac{\pi}{3}\mright)=\frac{1}{2} \\ \cos (\frac{3\pi}{4})=\frac{-\sqrt[]{2}}{2} \end{gathered}[/tex]By substituting the above trivial identities, we will have
[tex]\begin{gathered} \tan (\frac{\pi}{3})+\cos (\frac{\pi}{3})\text{.}\cos (\frac{3\pi}{4}) \\ \sqrt[]{3}+\frac{1}{2}\times\frac{-\sqrt[]{2}}{2} \\ =\sqrt[]{3}-\frac{\sqrt[]{2}}{4} \\ =\frac{4\sqrt[]{3}-\sqrt[]{2}}{4} \end{gathered}[/tex]Hence,
The SECOND OPTION is the right answer
[tex]\frac{4\sqrt[]{3}-\sqrt[]{2}}{4}[/tex]