Question

If $E=(\sin {10}^0+\cos {10}^0{)}^2+(\cos {10}^0-\sin {10}^0{)}^2$, then the value of ${\log }_{0.5}E$ will be

• A. $-1$
• B. $1$
• C. $0$
• D. $2$

Answer 1

The correct option is A $-1$

$E=(\sin {10}^{\circ }+\cos {10}^{\circ }{)}^2+(\cos {10}^{\circ }-\sin {10}^{\circ }{)}^2={\sin }^2{10}^{\circ }+{\cos }^2{10}^{\circ }+2\sin {10}^{\circ }\cos {10}^{\circ }+{\sin }^2{10}^{\circ }+{\cos }^2{10}^{\circ }-2\sin {10}^{\circ }\cos {10}^{\circ }=2\therefore {\log }_{0.5}E={\log }_{0.5}2={\log }_{2^{-1}}2=-{\log }_{2}2=-1,[\because {\log }_{a}a=1]$