Chapter 11

$$ \text { If } a \tan \alpha+b \cot 2 \alpha=c, a \cot \alpha-b \tan 2 \alpha=c, \text { eliminate } \alpha \text { . } $$

$$ \text { If } \frac{\cos x}{a}=\frac{\cos (x+y)}{b}=\frac{\cos (x+2 y)}{c}=\frac{\cos (x+3 y)}{d}, \text { prove that } b(b+d)=c(c+a) $$

$$ \text { If } a \cot ^{2} \alpha+b \cot ^{2} \beta=1, a \cos ^{2} \alpha+b \cos ^{2} \beta=1 \text { and } a \sin \alpha=b \sin \beta, \text { then prove that }\left(a^{2}-b^{2}\right)^{2}+a b=0 \text { . } $$

$$ \text { Let } \cos \alpha=\cos \beta \cos \phi=\cos \gamma \cos \theta \text { and } \sin \alpha=2 \sin \frac{\phi}{2} \sin \frac{\theta}{2}, \text { prove that } \tan ^{2} \frac{\alpha}{2}=\tan ^{2} \frac{\beta}{2} \tan ^{2} \frac{\gamma}{2} \text { . } $$

$$ \begin{aligned} &\text { If } \cos \theta=\cos \alpha \cos \beta \text { and } \cos \phi=\cos \gamma \cos \beta \text { where } \cos \beta \neq 0 \text { and } \tan \frac{p}{2}=\tan \frac{\sigma}{2} \tan \frac{\varphi}{2}, \text { prove that }\\\ &\sin ^{2} \beta=(\sec \alpha-1)(\sec \gamma-1) \end{aligned} $$

$$ \text { Find the value of } \left.\tan \frac{2 \pi}{5}-\tan \frac{\pi}{15}-\sqrt{3} \tan \frac{2 \pi}{5} \tan \frac{\pi}{15} \text { . \\{Ans. } \sqrt{3}\right\\} $$

$$ \text { If } f(x)=\cos (\log x) \text { , then evaluate } f(x) f(y)-\frac{1}{2}\left[f\left(\frac{x}{y}\right)+f(x y)\right] \text { . } $$

$$ \text { If } \cos \alpha+\cos \beta=0=\sin \alpha+\sin \beta, \text { then show that } \cos 2 \alpha+\cos 2 \beta=-2 \cos (\alpha+\beta) \text { . } $$

$$ \text { If } \sec \theta+\tan \theta=p \text { , obtain the values of } \sec \theta, \tan \theta \text { and } \sin \theta \text { in terms of } p \text { . } $$

$$ \text { If } \sin x+\sin ^{2} x=1, \text { show that } \cos ^{2} x+\cos ^{4} x=1 $$

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

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

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

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

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

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

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

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

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

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

$$ \text { If } m \cos (\theta+\alpha)=n \cos (\theta-\alpha), \text { show that }(m-n) \cot \theta=(m+n) \tan \alpha \text { . } $$

$$ \text { If } \cot ^{2} \theta=\cot (\theta-\alpha) \cot (\theta-\beta), \text { show that } \cot 2 \theta=\frac{1}{2}(\cot \alpha+\cot \beta) \text { . } $$

$$ \text { If } \tan \left(\frac{\pi}{4}+\frac{y}{2}\right)=\tan ^{3}\left(\frac{\pi}{4}+\frac{x}{2}\right), \text { prove that } \sin y=\frac{\sin x\left(3+\sin ^{2} x\right)}{1+3 \sin ^{2} x} \text { . } $$

$$ \text { If } \frac{\sin (\theta+A)}{\sin (\theta+B)}=\sqrt{\frac{\sin 2 A}{\sin 2 B}} \text { , prove that } \tan ^{2} \theta=\tan A \tan B \text { . } $$

If \(0<\alpha, \beta, \gamma<\pi\), prove that i. \(\quad \sin \alpha+\sin \beta+\sin \gamma>\sin (\alpha+\beta+\gamma)\) ii. \(\sin \alpha+\sin \beta+\sin \gamma>3 \sin \alpha \sin \beta \sin \gamma\)

\text { If } \cos x=\tan y, \cos y=\tan z, \cos z=\tan x, \text { prove that } \sin x=\sin y=\sin z=2 \sin 18^{\circ}

$$ \text { If } \sqrt{2} \cos A=\cos B+\cos ^{3} B \text { and } \sqrt{2} \sin A=\sin B-\sin ^{3} B, \text { show that } \sin (A-B)=\pm \frac{1}{3} \text { . } $$

$$ \text { If } \frac{\sin ^{4} A}{a}+\frac{\cos ^{4} A}{b}=\frac{1}{a+b}, \text { prove that } \frac{\sin ^{8} A}{a^{3}}+\frac{\cos ^{8} A}{b^{3}}=\frac{1}{(a+b)^{3}} $$

$$ \text { If } \sin (y+z-x), \sin (z+x-y), \sin (x+y-z) \text { be in A.P., prove that } \tan x, \tan y, \tan z \text { are also in A.P. } $$

$$ \text { If } \sec (\phi-\alpha), \sec \phi, \sec (\phi+\alpha) \text { be in A.P. prove that } \cos \phi=\sqrt{2} \cos \frac{\alpha}{2} \text { . } $$

If \(\sin x+\sin y=a, \cos x+\cos y=b\), show that i. \(\cos (x-y)=\frac{a^{2}+b^{2}-2}{2}\) ii. \(\tan \frac{x-y}{2}=\pm \sqrt{\frac{4-a^{2}-b^{2}}{a^{2}+b^{2}}}\).

$$ \text { If } \cos (\alpha+\beta)=\frac{4}{5} \text { and } \sin (\alpha-\beta)=\frac{5}{13} \text { and } \alpha, \beta \text { lie between } 0 \text { and } \frac{\pi}{4} \text { , find } \tan 2 \alpha \text { . } $$

$$ \text { If } \tan A-\tan B=x \text { and } \cot B-\cot A=y, \text { prove that } \cot (A-B)=\frac{1}{x}+\frac{1}{y} \text { . } $$

$$ \text { Prove that } \sin ^{2}(\theta+\alpha)+\sin ^{2}(\theta+\beta)-2 \cos (\alpha-\beta) \sin (\theta+\alpha) \sin (\theta+\beta) \text { is independent of } \theta \text { . } $$

$$ \text { If } \frac{\cos ^{4} x}{\cos ^{2} y}+\frac{\sin ^{4} x}{\sin ^{2} y}=1, \text { prove that } \frac{\cos ^{4} y}{\cos ^{2} x}+\frac{\sin ^{4} y}{\sin ^{2} x}=1 \text { . } $$

$$ \text { If } \sin x+\sin y=3(\cos y-\cos x), \text { prove that } \sin 3 x+\sin 3 y=0 $$

$$ \begin{aligned} &\text { If } x=\sin \theta(1+\sin \theta)+\cos \theta(1+\cos \theta) \text { and } y=\sin \theta(1-\sin \theta)+\cos \theta(1-\cos \theta), \text { prove that }\\\ &x^{2}-2 x-\sin 2 \theta=y^{2}+2 y-\sin 2 \theta=0 \text { . } \end{aligned} $$

$$ \text { If } \cos (A+B+C)=\cos A \cos B \cos C, \text { show that } 8 \sin (B+C) \sin (C+A) \sin (A+B)=-\sin 2 A \sin 2 B \sin 2 C \text { . } $$

$$ \text { If } \cos A+\cos B+\cos C=0, \text { prove that } \cos 3 A+\cos 3 B+\cos 3 C=12 \cos A \cos B \cos C $$

$$ \text { If } \tan \beta=\cos \theta \tan \alpha, \text { then prove that } \sin (\alpha-\beta)=\tan ^{2} \frac{\theta}{2} \sin (\alpha+\beta) $$

$$ \text { If } 2 \tan \beta+\cot \beta=\tan \alpha, \text { then } \cot \beta=2 \tan (\alpha-\beta) $$

$$ \text { If } \sin \theta=n \sin (\theta+2 \alpha), \text { show that } \tan (\theta+\alpha)=\frac{1+n}{1} \tan \alpha . $$

If an angle \(\theta\) be divided into two parts such that the tangent of one part is \(m\) times the tangent of the other, then prove that their difference \(\phi\) is obtained by the equation \(\sin \phi=\frac{m-1}{m+1} \sin \theta\).

$$ \text { If } x^{2} \sin ^{2}(\alpha+\beta)=\sin ^{2} \alpha+\sin ^{2} \beta-2 \sin \alpha \sin \beta \cos (\alpha-\beta), \text { show that } \tan \alpha=\frac{1 \pm x}{1 \mp x} \tan \beta \text { . } $$

$$ \text { If } \cos A=m \cos B, \text { then prove that } \cot \left(\frac{A+B}{2}\right)=\left(\frac{m+1}{m-1}\right) \tan \left(\frac{B-A}{2}\right) \text { . } $$

$$ \text { If } \sec (x+y)+\sec (x-y)=2 \sec x, \text { then prove that } \cos x=\pm \sqrt{2} \cos \frac{y}{2} \text { . } $$

$$ \text { If } \tan y=\frac{n \sin x \cos x}{1-n \sin ^{2} x}, \text { prove that } \tan (x-y)=(1-n) \tan x . $$

$$ \text { If } \sec \alpha \sec \beta+\tan \alpha \tan \beta=\tan \theta, \text { prove that } \cos 2 \theta \leq 0 $$

$$ \text { Suppose } \sin ^{3} x \sin 3 x=\sum_{m=0}^{n} C_{m} \cos ^{m} x, \text { is an identity in } x, \text { where } C_{0}, C_{1}, C_{2}, \ldots \ldots \ldots C_{n} \text { are real constants and } $$ $$ C_{n} \neq 0 \text { . Find the value of } n \text { . Also find } C_{0}, C_{1}, C_{2}, \ldots \ldots \ldots C_{n} \text { . } $$

$$ \text { If } \alpha=\frac{2 \pi}{7} \text { , prove that } \tan \alpha \tan 2 \alpha+\tan 2 \alpha \tan 4 \alpha+\tan 4 \alpha \tan \alpha=-7 $$