Question

Write the value of the expression $\frac{1-4\sin 10°\sin 70°}{2\sin 10°}$.

Answer 1

$$\begin{gathered} \frac{1-4\sin 10°\sin 70°}{2\sin 10°} \\ =\frac{1-2\left[2\sin 10°\sin 70°\right]}{2\sin 10°} \\ =\frac{1-2\left[\cos \left(10°-70°\right)-\cos \left(10°+70°\right)\right]}{2\sin 10°} \\ =\frac{1-2\left[\cos \left(-60°\right)-\cos 80°\right]}{2\sin 10°} \\ =\frac{1-2\left[\cos 60°-\cos 80°\right]}{2\sin 10°} \\ =\frac{1-2\left[{\displaystyle \frac{1}{2}}-\cos \left(90°-10°\right)\right]}{2\sin 10°} \\ =\frac{1-2\times {\displaystyle \frac{1}{2}}+2\cos \left(90°-10°\right)}{2\sin 10°} \\ =\frac{2\sin 10°}{2\sin 10°} \\ =1 \end{gathered}$$