Is sin(pi/3+x)-sin(pi/3-x)=sin(x) ?

The answer to whether sin(pi/3+x)-sin(pi/3-x)=sin(x) is True

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prove sin(pi/3+x)-sin(pi/3-x)=sin(x)

prove

sin (\frac{π}{3} + x) - sin (\frac{π}{3} - x) = sin (x)

True

Solution steps

sin (\frac{π}{3} + x) - sin (\frac{π}{3} - x) = sin (x)

Manipulating left side

sin (\frac{π}{3} + x) - sin (\frac{π}{3} - x)

Rewrite using trig identities

= sin (\frac{π}{3} + x) - (\frac{3}{2} cos (x) - \frac{1}{2} sin (x))

Rewrite using trig identities

= \frac{3}{2} cos (x) + \frac{1}{2} sin (x) - (\frac{3}{2} cos (x) - \frac{1}{2} sin (x))

Simplify

\frac{3}{2} cos (x) + \frac{1}{2} sin (x) - (\frac{3}{2} cos (x) - \frac{1}{2} sin (x)) : sin (x)

= sin (x)

We showed that the two sides could take the same form

\Rightarrow True

1 + cos (2 x) = 2 cos^{2} (x)

tan (x) (sin (x) + cot (x) cos (x)) = sec (x)

sin (x) cos^{2} (x) + sin^{3} (x) = sin (x)

\frac{csc ( x ) + sec ( x )}{tan ( x ) + 1} = csc (x)

cos (\frac{π}{3}) = \frac{1}{2}

Is sin(pi/3+x)-sin(pi/3-x)=sin(x) ?
The answer to whether sin(pi/3+x)-sin(pi/3-x)=sin(x) is True