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

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

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

prove

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

True

Solution steps

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

Manipulating left side

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

Rewrite using trig identities

= sin (\frac{π}{6} + 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

1 + tan^{2} (θ) = \frac{1}{cos ( θ )}

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

sin (x) + 5 \cdot cos (x) = 1

sin (2 x) - cos (2 x) = tan (2 x) - 1

0 = 0 (2) cos (θ)

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