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

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

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

prove

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

True

Solution steps

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

Manipulating left side

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

Rewrite using trig identities

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

3 cos (x + y) + 3 cos (x - y) = 6 cos (x) cos (y)

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

tan (x + \frac{π}{4}) = \frac{tan ( x ) + 1}{1 - tan ( x )}

csc (θ) cos (θ) tan (θ) = 1

sin (θ) cos (φ) = \frac{sin ( θ + φ ) + sin ( θ - φ )}{2}

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