Originally Posted by

**MarkFL2** Another approach would be to essentially use a linear combination identity as follows:

$\displaystyle \sqrt{2}\left(\sin\left(\frac{\pi}{4} \right)\cos\left(\theta+\frac{\pi}{6} \right)-\cos\left(\frac{\pi}{4} \right)\sin\left(\theta+\frac{\pi}{6} \right) \right)$

Now, within the brackets, use the angle-difference identity for sine to write:

$\displaystyle \sqrt{2}\sin\left(\frac{\pi}{4}-\left(\theta+\frac{\pi}{6} \right) \right)=\sqrt{2}\sin\left(\frac{\pi}{12}-\theta \right)$

Now, if we prefer, we may use the identity $\displaystyle \sin(\pi-x)=\sin(x)$ to write this as:

$\displaystyle \sqrt{2}\sin\left(\pi-\left(\frac{\pi}{12}-\theta \right \right)=\sqrt{2}\sin\left(\theta+\frac{11\pi}{12} \right)$