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

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

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

prove

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

True

Solution steps

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

Manipulating left side

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

Rewrite using trig identities

= cos (\frac{π}{3} + x) + \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

sec^{2} (x) + tan^{2} (x) = 2 sec^{2} (x) - 1

cos^{2} (2 x) = \frac{1 + cos ( 4 x )}{2}

sin^{2} (θ) csc^{2} (θ) - sin^{2} (θ) = cos^{2} (θ)

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

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

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