Question

If $x^3-1=0$,then $\sin \{({\omega }^{101}+{\omega }^{100})\pi +\frac{5\pi }{4}\}+\cos \{({\omega }^{1001}+{\omega }^{1000})\pi -\frac{\pi }{4}\}$ equals

• A. $\frac{2-\sqrt{2}}{2}$
• B. 0
• C. $\frac{2+\sqrt{2}}{2}$
• D. $\frac{-2+\sqrt{2}}{2}$

Answer 1

Answer: (B)

0

Let $w$ be the cube root of unity.
$\therefore w^3=1\&1+w+w^2=0$where $w=\frac{-1+i\sqrt{3}}{2}\&w^2=\frac{-1-i\sqrt{3}}{2}$
$z=\sin \{(w^{101}+w^{100})\pi +\frac{5\pi }{4}\}+\cos \{(w^{1001}+w^{1000})\pi -\frac{\pi }{4}\}\Rightarrow z=\sin \{{\left(w^3\right)}^{33}(w^2+w)\pi +\frac{5\pi }{4}\}+\cos \{{\left(w^3\right)}^{333}(w^2+w)\pi -\frac{\pi }{4}\}=\sin \left(\frac{\pi }{4}\right)+\cos \left(\frac{5\pi }{4}\right)\Rightarrow z=\frac{1}{\sqrt{2}}-\frac{1}{\sqrt{2}}=0$
Ans: B