Chapter 11

$$ \text { If } 4 n \alpha=\pi, \text { prove that } \tan \alpha \tan 2 \alpha \tan 3 \alpha \ldots \ldots \ldots . \tan (2 n-2) \alpha \tan (2 n-1) \alpha=1 \text { . } $$

$$ \begin{aligned} &\text { Prove that } x^{2}-x \cos (A+B)+1 \text { is a factor of }\\\ &2 x^{4}+4 x^{3} \sin A \sin B-x^{2}(\cos 2 A+\cos 2 B)+4 x \cos A \cos B-2 \text { . Also find the other factor. } \end{aligned} $$

$$ \text { Show that } 3\left(\sin ^{4}\left(\frac{3 \pi}{2}-x\right)+\sin ^{4}(3 \pi+x)\right)-2\left(\sin ^{6}\left(\frac{\pi}{2}+x\right)+\sin ^{6}(5 \pi-x)\right) \text { is independent of } x \text { . } $$

If \(\cos \alpha+\cos \beta+\cos \gamma=\sin \alpha+\sin \beta+\sin \gamma=0\). Prove that i. \(\quad \cot \frac{\alpha+\beta}{2}=\cot \gamma\) ii. \(\cos \frac{\alpha-\beta}{2}=\pm \frac{1}{2}\). iii. \(\cos ^{2} \alpha+\cos ^{2} \beta+\cos ^{2} \gamma=\sin ^{2} \alpha+\sin ^{2} \beta+\sin ^{2} \gamma=\frac{3}{2}\).