The value of cos (45 + θ) - sin (45° + θ) is:

Correct option is A

​$$\begin{aligned}\cos(45^{\circ}+\theta) &= \cos 45^{\circ}\cos\theta - \sin 45^{\circ}\sin\theta \\&= \frac{1}{\sqrt{2}}(\cos\theta - \sin\theta) \\\\\sin(45^{\circ}+\theta) &= \sin 45^{\circ}\cos\theta + \cos 45^{\circ}\sin\theta \\&= \frac{1}{\sqrt{2}}(\cos\theta + \sin\theta) \\\\\cos(45^{\circ}+\theta) - \sin(45^{\circ}+\theta) &= \frac{1}{\sqrt{2}}(\cos\theta - \sin\theta) - \frac{1}{\sqrt{2}}(\cos\theta + \sin\theta) \\&= \frac{1}{\sqrt{2}}[(\cos\theta - \sin\theta) - (\cos\theta + \sin\theta)] \\&= \frac{1}{\sqrt{2}}[\cos\theta - \sin\theta - \cos\theta - \sin\theta] \\&= \frac{1}{\sqrt{2}}(-2\sin\theta) \\&= -\sqrt{2}\sin\theta\end{aligned}$$​​