I have been trying to find a way to reduce following trigonometric statements to the reduced form below, but without succes. I haven't been able to grasp the typical train of thought I presume I would start off with before tackling the following two. These two steps occurred in two separate exercises on integrals and polar curves.
1) I would like to reduce the left side to the right:
$$\cos^{8}\left(\frac{\theta}{4}\right)+\cos^{6}\left(\frac{\theta}{4}\right)\cdot \sin^{2}\left(\frac{\theta}{4}\right) = \cos^{6}\left(\frac{\theta}{4}\right).$$
2) I would like to reduce the left side to the right: $$\sin\left(\frac{\theta}{4}\right)\cdot \cos(\theta)+\cos\left(\frac{\theta}{4}\right)\cdot \sin(\theta)=\sin\left(\frac{5}{4}\theta\right).$$
1) Using the Pythagorean identity:
$$\cos^{8}\left(\frac{\theta}{4}\right)+\cos^{6}\left(\frac{\theta}{4}\right)\cdot \sin^{2}\left(\frac{\theta}{4}\right)$$
$$=\cos^{6}\left(\frac{\theta}{4}\right)\cdot\cos^{2}\left(\frac{\theta}{4}\right)+\cos^{6}\left(\frac{\theta}{4}\right)\cdot\sin^{2}\left(\frac{\theta}{4}\right)$$
$$=\cos^{6}\left(\frac{\theta}{4}\right)\cdot\left[\cos^{2}\left(\frac{\theta}{4}\right)+\sin^{2}\left(\frac{\theta}{4}\right)\right]$$ $$=\cos^{6}\left(\frac{\theta}{4}\right)$$
2) Using sum- and addition formula
$\sin\left(\alpha+\beta\right)= \sin\left(\alpha\right)\cos\left(\beta\right)+\sin\left(\beta\right)\cos\left(\alpha\right)$ :
$$\sin\left(\frac{\theta}{4}\right)\cdot \cos(\theta)+\sin\left(\theta\right)\cdot\cos\left(\frac{\theta}{4}\right) $$
$$=\sin\left(\frac{5\theta}{4}\right)$$
Thanks to Gerry Myerson and John in the comments above.