Prove that: 2 5 12 12 = 3+2 2 - Vidyadip

Question

Prove that: $2\sin\frac{5\pi}{12}\cos\frac{\pi}{12}=\frac{\sqrt3+2}{2}$

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Answer

$\text{LHS}=2\sin\frac{5\pi}{12}\cos\frac{\pi}{12}$ $\because\ 2\sin\text{A}\sin\text{B}=\sin(\text{A+B})+\sin(\text{A}-\text{B})$ $=\ \Big(\frac{5\pi}{12}+\frac{\pi}{12}\Big)+\sin\Big(\frac{5\pi}{12}-\frac{\pi}{12}\Big)$ $=\ \sin\frac{\pi}{2}+\sin\frac{\pi}{3}$ $=\ 1+\frac{\sqrt3}{2}=\frac{2+\sqrt3}{2}=\text{RHS}(\text{Taking LCM})$

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