Is sin(pi/3+x)-cos(pi/6+x)=sin(x) ?

The answer to whether sin(pi/3+x)-cos(pi/6+x)=sin(x) is True

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prove sin(pi/3+x)-cos(pi/6+x)=sin(x)

prove

sin (\frac{π}{3} + x) - cos (\frac{π}{6} + x) = sin (x)

True

Solution steps

sin (\frac{π}{3} + x) - cos (\frac{π}{6} + x) = sin (x)

Manipulating left side

sin (\frac{π}{3} + x) - cos (\frac{π}{6} + 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

csc (2 x) - cot (2 x) = tan (x)

sin^{2} (x) (cot^{2} (x) + 1) = 1

tan^{2} (θ) = csc^{2} (θ) tan^{2} (θ) - 1

cos (x + \frac{π}{6}) + sin (x - \frac{π}{3}) = 0

tanh^{2} (x) + sech^{2} (x) = 1

Is sin(pi/3+x)-cos(pi/6+x)=sin(x) ?
The answer to whether sin(pi/3+x)-cos(pi/6+x)=sin(x) is True