Chapter 11

$$ \text { If } \alpha+\beta+\gamma=0, \text { prove that } \sin 2 \alpha+\sin 2 \beta+\sin 2 \gamma=2(\sin \alpha+\sin \beta+\sin \gamma)(1+\cos \alpha+\cos \beta+\cos \gamma) $$

If \(A+B=C\), prove that i. \(\quad \cos ^{2} A+\cos ^{2} B+\cos ^{2} C=1+2 \cos A \cos B \cos C\). ii. \(\quad \tan A \tan B \tan C=\tan C-\tan B-\tan A\).

\text { If } A+C=2 B, \text { prove that } \cot B=\frac{\sin A-\sin C}{\cos C-\cos A} \text { . }

If \(\alpha+\beta+\gamma+\delta=2 \pi\), prove that \(\cos \alpha+\cos \beta+\cos \gamma+\cos \delta+4 \cos \frac{\alpha+\beta}{2} \cos \frac{\alpha+\gamma}{2} \cos \frac{\alpha+\delta}{2}=0\) ii. \(\quad \sin \alpha-\sin \beta+\sin \gamma-\sin \delta+4 \cos \frac{\alpha+\beta}{2} \sin \frac{\alpha+\gamma}{2} \cos \frac{\alpha+\delta}{2}=0\) iii. \(\tan \alpha+\tan \beta+\tan \gamma+\tan \delta=\tan \alpha \tan \beta \tan \gamma \tan \delta(\cot \alpha+\cot \beta+\cot \gamma+\cot \delta)\).

\cos \frac{2 \pi}{7}+\cos \frac{4 \pi}{7}+\cos \frac{6 \pi}{7}=-\frac{1}{2}

$$ \cos \frac{\pi}{11}+\cos \frac{3 \pi}{11}+\cos \frac{5 \pi}{11}+\cos \frac{7 \pi}{11}+\cos \frac{9 \pi}{11}=\frac{1}{2} $$

$$ \cos 0^{\circ}+\cos \frac{\pi}{7}+\cos \frac{2 \pi}{7}+\cos \frac{3 \pi}{7}+\cos \frac{4 \pi}{7}+\cos \frac{5 \pi}{7}+\cos \frac{6 \pi}{7}=1 $$

$$ \sin \frac{2 \pi}{7}+\sin \frac{4 \pi}{7}+\sin \frac{6 \pi}{7}+\sin \frac{8 \pi}{7}+\sin \frac{10 \pi}{7}+\sin \frac{12 \pi}{7}=0 $$

$$ \text { If } n \text { is an integer greater than } 2, \text { prove that } \cos \frac{2 \pi}{n}+\cos \frac{4 \pi}{n}+\cos \frac{6 \pi}{n}+\ldots \ldots \ldots \text { to } n \text { terms }=0 \text { . } $$

$$ \text { Prove that } \sin ^{2} \alpha+\sin ^{2} 2 \alpha+\sin ^{2} 3 \alpha+\ldots \ldots \text { to } n \text { terms }=\frac{1}{4}[(2 n+1) \sin \alpha-\sin (2 n+1) \alpha] \operatorname{cosec} \alpha $$

$$ \begin{aligned} &\text { Prove that }\\\ &\sin ^{3} \alpha+\sin ^{3} 2 \alpha+\sin ^{3} 3 \alpha+\ldots \ldots \text { to n terms }\\\ &=\frac{3}{4} \sin \frac{(n+1) \alpha}{2} \sin \frac{n \alpha}{2} \operatorname{cosec} \frac{\alpha}{2}-\frac{1}{4} \sin \frac{3(n+1) \alpha}{2} \sin \frac{3 n \alpha}{2} \operatorname{cosec} \frac{3 \alpha}{2} \end{aligned} $$

$$ \begin{aligned} &\text { Prove that the sum of the product of sines of the angles } \alpha, 2 \alpha, 3 \alpha, \ldots ., n \alpha \text { taking two at a time is }\\\ &\frac{1}{2}\left[\frac{\sin ^{2} \frac{(n+1) \alpha}{2} \sin ^{2} \frac{n \alpha}{2}}{\sin ^{2} \frac{\alpha}{2}}+\frac{\cos (n+1) \alpha \sin n \alpha}{2 \sin \alpha}-\frac{n}{2}\right] . \end{aligned} $$

$$ \cos 20^{\circ} \cos 40^{\circ} \cos 60^{\circ} \cos 80^{\circ}=\frac{1}{16} $$

$$ 16 \cos \frac{2 \pi}{15} \cos \frac{4 \pi}{15} \cos \frac{8 \pi}{15} \cos \frac{14 \pi}{15}=1 $$

$$ \cos \frac{\pi}{65} \cos \frac{2 \pi}{65} \cos \frac{4 \pi}{65} \cos \frac{8 \pi}{65} \cos \frac{16 \pi}{65} \cos \frac{32 \pi}{65}=\frac{1}{64} $$

$$ \text { If } \theta=\frac{\pi}{2^{n}-1}, \text { prove that } \cos \theta \cos 2 \theta \cos 2^{2} \theta \ldots \ldots \ldots \ldots \cos 2^{n-1} \theta=-\frac{1}{2^{n}} $$

$$ \text { If } \theta=\frac{\pi}{2^{n}+1}, \text { prove that } \cos \theta \cos 2 \theta \cos 2^{2} \theta \ldots \ldots \ldots \ldots \cos 2^{n-1} \theta=\frac{1}{2^{n}} $$

$$ \cos \frac{\pi}{15} \cos \frac{2 \pi}{15} \cos \frac{3 \pi}{15} \cos \frac{4 \pi}{15} \cos \frac{5 \pi}{15} \cos \frac{6 \pi}{15} \cos \frac{7 \pi}{15}=\frac{1}{2^{7}} $$

$$ \cos \frac{\pi}{20} \cos \frac{3 \pi}{20} \cos \frac{7 \pi}{20} \cos \frac{9 \pi}{20}+\cos \frac{\pi}{15} \cos \frac{2 \pi}{15} \cos \frac{4 \pi}{15} \cos \frac{8 \pi}{15}=0 $$

$$ \begin{aligned} &(1+\sec 2 \theta)(1+\sec 4 \theta)(1+\sec 8 \theta) \ldots \ldots \ldots \ldots\left(1+\sec 2^{n} \theta\right)=\tan 2^{n} \theta \cot \theta \\ &\tan 20^{\circ} \tan 40^{\circ} \tan 80^{\circ}-\sqrt{3} \end{aligned} $$

$$ \tan 20^{\circ} \tan 40^{\circ} \tan 80^{\circ}=\sqrt{3} $$

$$ \sin \frac{\pi}{14} \sin \frac{3 \pi}{14} \sin \frac{5 \pi}{14}=\frac{1}{8} $$

$$ \sin \frac{\pi}{14} \sin \frac{3 \pi}{14} \sin \frac{5 \pi}{14} \sin \frac{7 \pi}{14} \sin \frac{9 \pi}{14} \sin \frac{11 \pi}{14} \sin \frac{13 \pi}{14}=\frac{1}{64} $$

$$ \begin{aligned} &\text { If } A=\frac{\pi}{2^{n+1}}, B=\frac{\pi}{2^{n+2}} \text { , then prove that }\\\ &(\cos A+\cos B)(\cos 2 A+\cos 2 B)\left(\cos 2^{2} A+\cos 2^{2} B\right) \ldots \ldots .\left(\cos 2^{n} A+\cos 2^{n} B\right)=\frac{1}{2^{n+1}}\left(\cos \frac{\pi}{2^{n+2}}-\cos \frac{\pi}{2^{n+1}}\right)^{-1} \end{aligned} $$

Prove that the roots of the equation \(8 x^{3}-4 x^{2}-4 x+1=0\) are \(\cos \frac{\pi}{7}, \cos \frac{3 \pi}{7}\) and \(\cos \frac{5 \pi}{7}\) and hence prove that i. \(\quad \cos \frac{\pi}{7}+\cos \frac{3 \pi}{7}+\cos \frac{5 \pi}{7}=\frac{1}{2}\) ii. \(\quad \cos \frac{\pi}{7} \cos \frac{3 \pi}{7}+\cos \frac{\pi}{7} \cos \frac{5 \pi}{7}+\cos \frac{3 \pi}{7} \cos \frac{5 \pi}{7}=-\frac{1}{2}\) iii. \(\cos \frac{\pi}{7} \cos \frac{3 \pi}{7} \cos \frac{5 \pi}{7}=-\frac{1}{8}\) iv. \(\left(1-\cos \frac{\pi}{7}\right)\left(1-\cos \frac{3 \pi}{7}\right)\left(1-\cos \frac{5 \pi}{7}\right)=\frac{1}{8}\) v. the equation whose roots are \(\cos ^{2} \frac{\pi}{7}, \cos ^{2} \frac{3 \pi}{7}\) and \(\cos ^{2} \frac{5 \pi}{7}\) is \(64 x^{3}-80 x^{2}+24 x-1=0\) vi. \(\quad \cos ^{2} \frac{\pi}{7}+\cos ^{2} \frac{3 \pi}{7}+\cos ^{2} \frac{5 \pi}{7}=\frac{5}{4}\) vii. the equation whose roots are \(\sec \frac{\pi}{7}, \sec \frac{3 \pi}{7}\) and \(\sec \frac{5 \pi}{7}\) is \(x^{3}-4 x^{2}-4 x+8=0\) viii. \(\sec \frac{\pi}{7}+\sec \frac{3 \pi}{7}+\sec \frac{5 \pi}{7}=4\) ix. the equation whose roots are \(\sec ^{2} \frac{\pi}{7}, \sec ^{2} \frac{3 \pi}{7}\) and \(\sec ^{2} \frac{5 \pi}{7}\) is \(x^{3}-24 x^{2}+80 x-64=0\) x. \(\quad \sec ^{2} \frac{\pi}{7}+\sec ^{2} \frac{3 \pi}{7}+\sec ^{2} \frac{5 \pi}{7}=24\) xi. the equation whose roots are \(\tan ^{2} \frac{\pi}{7}, \tan ^{2} \frac{3 \pi}{7}\) and \(\tan ^{2} \frac{5 \pi}{7}\) is \(x^{3}-21 x^{2}+35 x-7=0\) xii. \(\tan ^{2} \frac{\pi}{7}+\tan ^{2} \frac{3 \pi}{7}+\tan ^{2} \frac{5 \pi}{7}=21\) xiii. \(\tan \frac{\pi}{7} \tan \frac{3 \pi}{7} \tan \frac{5 \pi}{7}=-\sqrt{7}\) xiv. the equation whose roots are \(\cot ^{2} \frac{\pi}{7}, \cot ^{2} \frac{3 \pi}{7}\) and \(\cot ^{2} \frac{5 \pi}{7}\) is \(7 x^{3}-35 x^{2}+21 x-1=0\) \(\mathrm{xv}, \quad \cot ^{2} \frac{\pi}{7}+\cot ^{2} \frac{3 \pi}{7}+\cot ^{2} \frac{5 \pi}{7}=5\)

$$ \text { Find the equation whose roots are } \left.\cos \frac{2 \pi}{7}, \cos \frac{4 \pi}{7} \text { and } \cos \frac{6 \pi}{7} \text { . Ans. } 8 x^{3}+4 x^{2}-4 x-1=0\right\\} $$

$$ \tan \frac{\pi}{7} \tan \frac{2 \pi}{7} \tan \frac{3 \pi}{7}=\sqrt{7} $$

$$ \left(\tan ^{2} \frac{\pi}{7}+\tan ^{2} \frac{2 \pi}{7}+\tan ^{2} \frac{3 \pi}{7}\right)\left(\cot ^{2} \frac{\pi}{7}+\cot ^{2} \frac{2 \pi}{7}+\cot ^{2} \frac{3 \pi}{7}\right)=105 $$

$$ \text { Find the value of } \sin ^{2} \frac{\pi}{7}+\sin ^{2} \frac{2 \pi}{7}+\sin ^{2} \frac{3 \pi}{7} \text { . } $$

$$ \sin \frac{\pi}{7} \sin \frac{2 \pi}{7} \sin \frac{3 \pi}{7}=\frac{\sqrt{7}}{8} $$

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

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

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

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

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

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

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

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

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

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

$$ \text { If } c \cos ^{3} \theta+3 c \cos \theta \sin ^{2} \theta=m \text { and } c \sin ^{3} \theta+3 c \cos ^{2} \theta \sin \theta=n, \text { prove that }(m+n)^{\frac{2}{3}}+(m-n)^{\frac{2}{3}}=2 c^{\frac{2}{3}} $$

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

$$ \text { If } \cos ^{2} \theta=\frac{1}{3}\left(a^{2}-1\right) \text { and } \tan ^{2} \frac{\theta}{2}=\tan ^{\frac{2}{3}} \alpha, \text { prove that } \cos ^{\frac{2}{3}} \alpha+\sin ^{\frac{2}{3}} \alpha=\left(\frac{2}{a}\right)^{\frac{2}{3}} \text { . } $$

$$ \begin{aligned} &\text { If } \frac{\tan (\theta+x)}{a}=\frac{\tan (\theta+y)}{b}=\frac{\tan (\theta+z)}{c} \text { prove that }\\\ &\frac{a+b}{a-b} \sin ^{2}(x-y)+\frac{b+c}{b-c} \sin ^{2}(y-z)+\frac{c+a}{c-a} \sin ^{2}(z-x)=0 \end{aligned} $$

$$ \text { If } a \sin x^{2}=b \cos x^{2}=\frac{2 c \tan x^{2}}{1-\tan ^{2} x^{2}}, \text { prove that }\left(a^{2}-b^{2}\right)^{2}=4 c^{2}\left(a^{2}+b^{2}\right) \text { . } $$

$$ \text { If } \cot \alpha=\left(x^{3}+x^{2}+x\right)^{\frac{1}{2}}, \cot \beta=\left(x+x^{-1}+1\right)^{\frac{1}{2}} \text { and } \cot \gamma=\left(x^{-3}+x^{-2}+x^{-1}\right)^{-\frac{1}{2}}, \text { prove that } \alpha+\beta=\gamma \text { . } $$

$$ \text { If } \frac{\cos ^{3} \theta}{\cos (\alpha-3 \theta)}=\frac{\sin ^{3} \theta}{\sin (\alpha-3 \theta)}=p, \text { show that } \cos \alpha=\frac{2 p^{2}-1}{p} $$

$$ \left.f \cot \theta+\tan \theta=x, \sec \theta-\cos \theta=y \text { , eliminate } \theta \text { . \\{Ans. }(x y)^{\frac{2}{3}}\left(x^{\frac{2}{3}}-y^{\frac{2}{3}}\right)=1\right\\} $$

$$ \text { If } \left.a \sec \theta=1-b \tan \theta, a^{2} \sec ^{2} \theta=5+b^{2} \tan ^{2} \theta, \text { eliminate } \theta \text { . \\{Ans. } a^{2} b^{2}+4 a^{2}-9 b^{2}=0\right\\} $$

$$ \text { If } a \sec \theta+b \tan \theta+c=0, p \sec \theta+q \tan \theta+r=0, \text { eliminate } \theta \text { . } $$