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

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

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

prove

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

True

Solution steps

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

Manipulating left side

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

Rewrite using trig identities

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

Rewrite using trig identities

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

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

= 0

We showed that the two sides could take the same form

\Rightarrow True

\frac{sin ( θ ) + cos ( θ )}{cos ( θ )} = 1 + tan (θ)

sec (x) (sec (x) - cos (x)) = tan^{2} (x)

\frac{1}{sin ( x )} - sin (x) = cot (x) cos (x)

sin (3 x) = 3 sin (x) - 4 (sin (x))^{3}

\frac{sin ( θ ) - 1}{cos ( θ )} = tan (θ) - sec (θ)

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