Question

The value of expression $\frac{1-4\sin {10}^{\circ }\sin {70}^{\circ }}{2\sin {10}^{\circ }}$ is

• A. $\frac{1}{2}$
• B. $1$
• C. $2$
• D. none of these

Answer 1

The correct option is B $1$

$\frac{1-4\sin 10\sin 70}{2\sin 10}$

$[2\sin A\sin B=\cos (A-B)-\cos (A+B\left)\right]$

$=\frac{1-2\left[\cos \right(10-70)-\cos (10+70\left)\right]}{2\sin 10}$

$\left[\cos \right(-\theta )=\cos \theta ]$

$=\frac{1-2\cos 60+2\cos 80}{2\sin 10}$

$=\frac{1-2\left(1/2\right)+2\cos 80}{2\sin 10}=\frac{\cos 80}{\sin 10}$

$\left[\cos \right(90-\theta )=\sin \theta ,\therefore \cos (90-10)=\sin 10]$

$=\frac{\sin 10}{\sin 10}=1$.