Answer
$$\cos(45^\circ-\theta)=\frac{\sqrt2(\cos\theta+\sin\theta)}{2}$$
Work Step by Step
$$X=\cos(45^\circ-\theta)$$
To expand the formula, cosine difference identity would be used:
$$\cos(A-B)=\cos A\cos B+\sin A\sin B$$
That means
$$X=\cos45^\circ\cos\theta+\sin45^\circ\sin\theta$$
$$X=\frac{\sqrt2}{2}\cos\theta+\frac{\sqrt2}{2}\sin\theta$$
$$X=\frac{\sqrt2\cos\theta+\sqrt2\sin\theta}{2}$$
$$X=\frac{\sqrt2(\cos\theta+\sin\theta)}{2}$$
Therefore, $$\cos(45^\circ-\theta)=\frac{\sqrt2(\cos\theta+\sin\theta)}{2}$$
