Question

Prove sin(5 pi/6+x)+sin(5pi/6-x)=cos x.​

Answer 1

$\underline{\textsf{To prove:}}$

$\mathsf{sin\left(\dfrac{5\pi}{6}+x\right)+sin\left(\dfrac{5\pi}{6}-x\right)=cosx}$

$\underline{\textsf{Solution:}}$

$\mathsf{Consider,}$

$\mathsf{sin\left(\dfrac{5\pi}{6}+x\right)+sin\left(\dfrac{5\pi}{6}-x\right)}$

$\mathsf{Using}$

$\boxed{\mathsf{sin(A+B)+sin(A-B)=2\,sinA\,cosB}}$

$\mathsf{=2\,sin\dfrac{5\pi}{6}\,cosx}$

$\mathsf{=2\,sin\left(\pi-\dfrac{\pi}{6}\right)\,cosx}$

$\mathsf{=2\,sin\dfrac{\pi}{6}\,cosx}$

$\mathsf{=2\left(\dfrac{1}{2}\right)\,cosx}$

$\mathsf{=cosx}$

$\therefore\boxed{\mathsf{sin\left(\dfrac{5\pi}{6}+x\right)+sin\left(\dfrac{5\pi}{6}-x\right)=cosx}}$

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