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

= cos (x + \frac{π}{6}) + \frac{1}{2} sin (x) - \frac{3}{2} cos (x)

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

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

\frac{cos ( θ ) sec ( θ )}{cot ( θ )} = tan (θ)

csc^{2} (u) - cos (u) sec (u) = cot^{2} (u)

\frac{cos ^{2} ( x )}{sin ( x )} + sin (x) = csc (x)

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

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