91Ó°ÊÓ

$$ \text { Find the value of } \operatorname{cosec}^{2} \frac{\pi}{7}+\cos e c^{2} \frac{2 \pi}{7}+\operatorname{cosec}^{2} \frac{3 \pi}{7} \text { . } $$

$$ \sin \frac{2 \pi}{7}+\sin \frac{4 \pi}{7}+\sin \frac{8 \pi}{7}=\frac{\sqrt{7}}{2} $$

$$ \text { Show that } \sin \frac{\pi}{14} \text { is a root of the equation } 8 x^{3}-4 x^{2}-4 x+1=0 \text { and find the other roots. } $$

$$ \text { If } \frac{\cos \alpha}{\cos \beta}=n, \frac{\sin \alpha}{\sin \beta}=m, \text { show that }\left(m^{2}-n^{2}\right) \sin ^{2} \beta=1-n^{2} \text { . } $$

$$ \text { If } \frac{a x}{\cos \theta}+\frac{b y}{\sin \theta}=a^{2}-b^{2} \text { and } \frac{a x \sin \theta}{\cos ^{2} \theta}-\frac{b y \cos \theta}{\sin ^{2} \theta}=0, \text { show that }(a x)^{\frac{2}{3}}+(b y)^{\frac{2}{3}}=\left(a^{2}-b^{2}\right)^{\frac{2}{3}} \text { . } $$

$$ \text { If } \sec \theta-\cos \theta=a \text { and } \operatorname{cosec} \theta-\sin \theta=b, \text { prove that } a^{2} b^{2}\left(a^{2}+b^{2}+3\right)=1 $$

$$ \text { If } \operatorname{cosec} \theta-\sin \theta=a^{3}, \sec \theta-\cos \theta=b^{3}, \text { prove that } a^{2} b^{2}\left(a^{2}+b^{2}\right)=1 $$

$$ \text { If } \tan \theta+\sin \theta=m \text { and } \tan \theta-\sin \theta=n, \text { prove that } m^{2}-n^{2}=\pm 4 \sqrt{m n} \text { . } $$

$$ \text { If } a \cos \theta+b \sin \theta=p \text { and } a \sin \theta-b \cos \theta=q, \text { prove that } a^{2}+b^{2}=p^{2}+q^{2} \text { . } $$

$$ \text { If } \cot \theta(1+\sin \theta)=4 m \text { and } \cot \theta(1-\sin \theta)=4 n, \text { prove that }\left(m^{2}-n^{2}\right)^{2}=m n \text { . } $$