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

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

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

prove

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

True

Solution steps

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

Manipulating left side

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

Rewrite using trig identities

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

Rewrite using trig identities

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

Simplify

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

= cos (x)

We showed that the two sides could take the same form

\Rightarrow True

coth^{2} (x) - 1 = csch^{2} (x)

\frac{cos ^{4} ( x ) - sin ^{4} ( x )}{cos ^{2} ( x )} = 1 - tan^{2} (x)

csc^{2} (x) + cot^{2} (x) = 2 csc^{2} (x) - 1

cot (\frac{π}{2} - x) = tan (x)

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

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