Prove That,sin 10°sin50° + sin50°sin250°+ sin250°sin10°=-3/4

sin 10°sin50° + sin50°sin250°+ sin250°sin10°=-3/4

Prove That,sin 10°sin50° + sin50°sin250°+ sin250°sin10°=-3/4

Solution:

sin 10°sin50° + sin50°sin250° + sin250°sin10° 

= $\frac{1}{2}$ (sin 10°sin50° + sin50°sin250° + sin250°sin10°)

= $\frac{1}{2}$(2sin 10°sin50° + 2sin50°sin250° + 2sin250°sin10°)

= $\frac{1}{2}$ {cos (10°-50°) – cos(10° + 50°) + cos(50° – 250°)$- cos(50° + 250°) + cos(250° – 10°) – cos(250° + 10°)}

= $\frac{1}{2}$(cos40° -cos60° + cos200° – cos300° + cos240° – cos260°)

= $\frac{1}{2}$ (cos40°+cos200°-cos260°-cos300°+cos240°-cos60°)

= $\frac{1}{2}$ { cos 40°+cos(180° + 20°) – cos(180° + 80°) – cos(360° – 60°) + cos(180° + 60°) – cos60°}

= $\frac{1}{2}$(cos 40°-cos20°+cos80°-cos60°-cos60°-cos60°)

= $\frac{1}{2}$(cos40° -cos20°+sin10°– $\frac{1}{2}$  -$\frac{1}{2}$  – $\frac{1}{2}$)

=$\frac{1}{2}$(2 $\sin \frac{40 + 20}{2}sin\frac{20 – 40}{2}$ +sin10° – $\frac{3}{2}$ )    

= $\frac{1}{2}${ 2 sin30°sin(-10°)+sin10°-$\frac{3}{2}$}

= $\frac{1}{2}$(-2× $\frac{1}{2}$ ×sin10°+sin10°-$\frac{3}{2}$)

= $\frac{1}{2}$ (-sin10°+sin10°-$\frac{3}{2}$)

= $- \frac{3}{4}$ [Proved] 

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