Answer
$$\cos(A+2\pi)=\cos A$$
$$\sin(A+2\pi)=\sin A$$
The results show that adding an angle of $2\pi$, which is the period of sine and cosine, to angle $A$ would not change its value of sine and cosine.
Work Step by Step
* Recall the addition formulas:
$$\cos(A+B)=\cos A\cos B-\sin A\sin B$$
$$\sin(A+B)=\sin A\cos B+\cos A\sin B$$
If we take $B=2\pi$, then we have:
1) The addition formula for cosine:
$$\cos(A+2\pi)=\cos A\cos(2\pi)-\sin A\sin(2\pi)$$
$$\cos(A+2\pi)=\cos A\times1-\sin A\times0$$
$$\cos(A+2\pi)=\cos A$$
2) The addition formula for sine:
$$\sin(A+2\pi)=\sin A\cos(2\pi)+\cos A\sin(2\pi)$$
$$\sin(A+2\pi)=\sin A\times1+\cos A\times0$$
$$\sin(A+2\pi)=\sin A$$
As we can see, if we add $2\pi$ to an angle, the value of its sine and cosine is unchanged. This is because $2\pi$ is exactly one period of cosine and sine. In other words, in the unit circle, adding an angle of $2\pi$ to an angle $A$ would lead you to the exact same spot, so the value of sine and cosine is still the same.
