91Ó°ÊÓ

$$ \text { If } \sin x+\sin ^{2} x=1, \text { then find the value of } \cos ^{8} x+2 \cos ^{6} x+\cos ^{4} x \text { . } $$

$$ \text { If } \cos \theta+\sin \theta=\sqrt{2} \cos \theta, \text { show that } \cos \theta-\sin \theta=\sqrt{2} \sin \theta $$

$$ \text { If } \cos 3 x=-\frac{3 \sqrt{6}}{8}, \text { show that the three values of } \cos x \text { are } \frac{\sqrt{6}}{2} \sin \frac{\pi}{6}, \frac{\sqrt{6}}{2} \sin \frac{\pi}{10},-\frac{\sqrt{6}}{2} \sin \frac{3 \pi}{10} \text { . } $$

$$ \text { If } 0<\alpha, \beta<\pi \text { and } \cos \alpha+\cos \beta-\cos (\alpha+\beta)=\frac{3}{2}, \text { prove that } \alpha=\beta=\frac{\pi}{3} \text { . } $$

$$ \text { If } a \cos \theta-b \sin \theta=c, \text { show that } a \sin \theta+b \cos \theta=\pm \sqrt{a^{2}+b^{2}-c^{2}} $$

$$ \text { If } 3 \sin \theta+5 \cos \theta=5, \text { show that } 5 \sin \theta-3 \cos \theta=3 \text { or }-3 \text { . } $$

$$ \begin{aligned} &\text { If }(1+\sin A)(1+\sin B)(1+\sin C)=(1-\sin A)(1-\sin B)(1-\sin C) \text { prove that each side is equal to }\\\ &\pm \cos A \cos B \cos C \end{aligned} $$

$$ \begin{aligned} &\text { Prove that }\left(\frac{\cos A+\cos B}{\sin A-\sin B}\right)^{n}+\left(\frac{\sin A+\sin B}{\cos A-\cos B}\right)^{n}=2 \cot ^{n}\left(\frac{A-B}{2}\right) \text { or } 0 \text { according as } n \text { is even or odd }\\\ &\text { positive integer } \end{aligned} $$

$$ \text { If } \cos (A+B) \sin (C+D)=\cos (A-B) \sin (C-D), \text { prove that } \cot A \cot B \cot C=\cot D \text { . } $$

$$ \text { If } m \tan \left(\theta-30^{\circ}\right)=n \tan \left(\theta+120^{\circ}\right), \text { show that } \cos 2 \theta=\frac{m+n}{2(m-n)} \text { . } $$