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JEE Advanced
Exam
JEE Advanced
Subject
Mathematics
Chapter
Trigonometry

JEE Advanced · Mathematics · Trigonometry

Let x= 4 80^ 65^ 55^ 20^ + 50^ + 70^ , then x is:

Let x=4sin⁡80∘sin⁡65∘cos⁡55∘sin⁡20∘+sin⁡50∘+sin⁡70∘x=\frac{4 \sin 80^{\circ} \sin 65^{\circ} \cos 55^{\circ}}{\sin 20^{\circ}+\sin 50^{\circ}+\sin 70^{\circ}}, then xx is:

Correct answer

1

Step-by-step solution

 Denominator =sin⁡20∘+2sin⁡60∘⋅cos⁡10∘=sin⁡20∘+3cos⁡10∘ and Numerator =2sin⁡80∘(sin⁡120∘+sin⁡10∘)=2sin⁡80∘32+cos⁡70∘−cos⁡90∘=3cos⁡10∘+sin⁡20∘∴x=1\begin{aligned} & \text { Denominator }=\sin 20^{\circ}+2 \sin 60^{\circ} \cdot \cos 10^{\circ}=\sin 20^{\circ}+\sqrt{3} \cos 10^{\circ} \\ & \text { and Numerator }=2 \sin 80^{\circ}\left(\sin 120^{\circ}+\sin 10^{\circ}\right) \\ & =2 \sin 80^{\circ} \frac{\sqrt{3}}{2}+\cos 70^{\circ}-\cos 90^{\circ}=\sqrt{3} \cos 10^{\circ}+\sin 20^{\circ} \\ & \therefore \quad x=1 \\ & \end{aligned}
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