Is cos(x-pi/6)-sin(x+pi/3)=0 ?

The answer to whether cos(x-pi/6)-sin(x+pi/3)=0 is True

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prove cos(x-pi/6)-sin(x+pi/3)=0

prove

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

True

Solution steps

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

Manipulating left side

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

Rewrite using trig identities

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

Rewrite using trig identities

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

\frac{3}{2} cos (x) + \frac{1}{2} sin (x) - (\frac{1}{2} sin (x) + \frac{3}{2} cos (x)) = 0

= 0

We showed that the two sides could take the same form

\Rightarrow True

cot^{2} (α) + 1 = \frac{1}{sin ^{2} ( α )}

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

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

\frac{cos ( 3 x ) - cos ( 5 x )}{sin ( 3 x ) + sin ( 5 x )} = tan (x)

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

Is cos(x-pi/6)-sin(x+pi/3)=0 ?
The answer to whether cos(x-pi/6)-sin(x+pi/3)=0 is True