Question

How does cos(pi/6-x)-cos(x+pi/6) become sin(x) ?

Answer 1

\[\cos a-\cos b=-2\sin(\frac{a+b}2)\sin(\frac{a-b}2)\]might help. not that I prove that identity

Answer 2

are you not satisfied?

Answer 3

I am still solving your equation, sorry :D

Answer 4

\[a=\frac\pi6-x;~~~b=x+\frac\pi6\]\[\frac{a+b}2=\frac{\frac\pi6-\cancel x+\frac\pi6+\cancel x}2=\frac{\frac\pi3}2=\frac\pi6\]\[\frac{a-b}2={\cancel{\frac\pi6}-x-(\cancel{\frac\pi6}+x)\over2}=-\frac{2x}2=-x\]

Answer 5

\[\cos a-\cos b=-2\sin(\frac{a+b}2)\sin(\frac{a-b}2)\]\[\cos(\frac\pi6-x)-\cos (x+\frac\pi6)=-2\sin(\frac\pi6)\sin(-x)\]

Answer 6

remembering that\[\sin\frac\pi6=\frac12\]and that\[\sin(-x)=-\sin x\]we get\[=-2(\frac12)(-\sin x)=\sin x\]

Answer 7

thanks so much for your clear explanation!