ISI2019-MMA-7

Question

The value of $\frac{1}{2\sin10^\circ}$ – $2\sin70^\circ$ is 

• A. $-1/2$ 
• B. $-1$
• C. $1/2$
• D. $1$

Comments

yes,Ans will be 1

Answer 1

$\displaystyle{\frac{1}{(2sin10^{\circ})}-2sin70^{\circ} = \frac{1}{(2sin10^{\circ})}-2sin(90^{\circ}-20^{\circ}) = \frac{1}{(2sin10^{\circ})}-2cos20^{\circ}}$

$\displaystyle{= \frac{1-4cos20^{\circ}sin10^{\circ}}{2sin10^{\circ}} = \frac{1-2(2cos20^{\circ}sin10^{\circ})}{(2sin10^{\circ})}}$

$\displaystyle{= \frac{1}{2sin10^{\circ}}[1-2\{sin(20^{\circ}+10^{\circ})-sin(20^{\circ}-10^{\circ})\}]}$

$= \displaystyle{\frac{1}{2sin10^{\circ}}[1-2{sin30^{\circ}+2sin10^{\circ}}] = \frac{1}{2sin10^{\circ}}[1-2\frac{1}{2}+2sin10^{\circ}]}$

$= \displaystyle{\frac{1}{2sin10^{\circ}}[1-1+2sin10^{\circ}] = \frac{1}{2sin10^{\circ}}[2sin10^{\circ}] =1}$