Answer
The statement $$\cos85^\circ\cos40^\circ+\sin85^\circ\sin40^\circ=\frac{\sqrt2}{2}$$ is true.
Work Step by Step
$$\cos85^\circ\cos40^\circ+\sin85^\circ\sin40^\circ=\frac{\sqrt2}{2}$$
We examine the left side:
$$X=\cos85^\circ\cos40^\circ+\sin85^\circ\sin40^\circ$$
Remember the cosine difference identity:
$$\cos(A-B)=\cos A\cos B+\sin A\sin B$$
So $X$ is in fact the right side of the above identity with $A=85^\circ$ and $B=40^\circ$.
Therefore, we can simplify $X$:
$$X=\cos(85^\circ-40^\circ)$$
$$X=\cos45^\circ$$
$$X=\frac{\sqrt2}{2}$$
That means the statement $$\cos85^\circ\cos40^\circ+\sin85^\circ\sin40^\circ=\frac{\sqrt2}{2}$$ is true.
