Chapter 1

Prove that the sum of \(\tan x \tan 2 x+\tan 2 x \tan 3 x+\ldots+\tan x \tan (n+1) x\) \(=\cot x \tan (n+1) x-(n+1)\)

If \(\sec x=p+\frac{1}{p}\), then \(\sec x+\tan x\) is (a) \(p\) (b) \(2 p\) (c) \(\frac{1}{4 p}\) (d) \(\frac{4}{p}\).

Find the values of the expression $$ \begin{aligned} &3(\sin x-\cos x)^{4}+6(\sin x+\cos x)^{2} \\ &+4\left(\sin ^{6} x+\cos ^{6} x\right) \end{aligned} $$

Prove that \(\operatorname{cosec} x+\operatorname{cosec} 2 x+\operatorname{cosec} 4 x+\ldots .\) to \(n\) terms \(=\cot \left(\frac{x}{2}\right)-\cot \left(2^{n-1} x\right)\)

If \(\operatorname{cosec} x-\sin x=a^{3}, \sec x-\cos x=b^{3}\), then \(a^{2} b^{2}\) \(\left(a^{2}+b^{2}\right)\) is (a) 0 (b) 1 (c) \(-1\) (d) \(a b\).

Prove that, \(\cot \left(16^{\circ}\right) \cot \left(44^{\circ}\right)+\cot \left(44^{\circ}\right) \cot \left(76^{\circ}\right)\) \(-\cot \left(76^{\circ}\right) \cot \left(16^{\circ}\right)=3\)

If \(\sec x+\cos x=2\), then the value of \(\sec ^{3} x\left(1+\sec ^{3} x\right)+\cos ^{3} x\left(1+\cos ^{3} x\right)\) is (a) 2 (b) 4 (c) 6 (d) 8 .

The value of \(\cos \left(\frac{2 \pi}{7}\right)+\cos \left(\frac{4 \pi}{7}\right)+\cos \left(\frac{6 \pi}{7}\right)\) is (a) \(1 / 2\) (b) \(-1 / 2\) (c) 0 (d) None.

Find the value of \(\tan \frac{\pi}{20} \cdot \tan \frac{3 \pi}{20} \cdot \tan \frac{5 \pi}{20}, \tan \frac{7 \pi}{20} \cdot \tan \frac{9 \pi}{20}\)

If \(\theta=\frac{\pi}{2^{n}-1}\), prove that \(2^{n} \cos \theta\) \(\cos (2 \theta) \cdot \cos (4 \theta) \cdot \cos (8 \theta) \ldots \cos \left(2^{n-1} \theta\right)=-1\)

Which of the following is smallest? (a) \(\sin 1\) (b) \(\sin 2\) (c) \(\sin 3\) (d) \(\sin 4\).

If \(x=\frac{2 \sin \theta}{1+\cos \theta+\sin \theta}\), then find the values of \(\frac{1-\cos \theta+\sin \theta}{1+\sin \theta}\)

Prove that \(\sin \left(\frac{2 \pi}{7}\right)+\sin \left(\frac{4 \pi}{7}\right)+\sin \left(\frac{8 \pi}{7}\right)=\frac{\sqrt{7}}{2}\)

Which of the following is greatest ? (a) \(\sin 1\) (b) \(\cos 1\) (c) \(\tan 1\) (d) \(\cot 1\)

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

Prove that \(\tan ^{2}\left(\frac{\pi}{16}\right)+\tan ^{2}\left(\frac{2 \pi}{16}\right)+\ldots+\tan ^{2}\left(\frac{7 \pi}{16}\right)=35\)

If \(A=\cos (\cos x)+\sin (\cos x)\), then the least and greatest value of \(A\) are (a) 0,2 (b) \(-1,1\) (c) \(-\sqrt{2}, \sqrt{2}\) (d) \(0, \sqrt{2}\).

Prove that, \(\left(\tan ^{2}\left(\frac{\pi}{7}\right)+\tan ^{2}\left(\frac{2 \pi}{7}\right)+\tan ^{2}\left(\frac{3 \pi}{7}\right)\right)\) \(\times\left(\cot ^{2}\left(\frac{\pi}{7}\right)+\cot ^{2}\left(\frac{2 \pi}{7}\right)+\cot ^{2}\left(\frac{3 \pi}{7}\right)\right)=105\)

If \(A+B=\frac{\pi}{3}, A, B>0\) then the maximum value of \(\tan A \cdot \tan B\) is (a) \(1 / 3\) (b) 1 (c) \(1 / 2\) (d) \(2 / 3\)

If \(\sin x+\sin ^{2} x+\sin ^{3} x=1\), then find the value of \(\cos ^{6} x-4 \cos ^{4} x+8 \cos ^{2} x\)

Prove that, \(\frac{3+\cot \left(76^{\circ}\right) \cot \left(16^{\circ}\right)}{\cot \left(76^{\circ}\right)+\cot \left(16^{\circ}\right)}=\cot \left(44^{\circ}\right)\)

The maximum value of a \(\sin 2 x+b \cos 2 x\) for all real \(x\) is (a) \(a+b\) (b) \(\sqrt{a^{2}+b^{2}}\) (c) \(\operatorname{Max}\\{|a|,|b|\\}\) (d) \(\operatorname{Max}\\{a, b\\}\)

If \(\cos x+\cos y+\cos \alpha=0=\sin x+\sin y+\sin \alpha\) then find the value of \(\cot \left(\frac{x+y}{2}\right)\).

If \(\cos x+\cos y+\cos z=0\), then prove that \(\cos (3 x)+\cos (3 y)+\cos (3 z)=12 \cos x \cos y \cos z\)

Prove that, \(\tan ^{6}\left(\frac{\pi}{9}\right)-33 \tan ^{4}\left(\frac{\pi}{9}\right)+27 \tan ^{2}\left(\frac{\pi}{9}\right)=3\)

If \(\sin x+\sin ^{2} x=1\), then find the value of \(\cos ^{2} x+\cos ^{4} x\).

If \(\cos A+\cos B+\cos C=0=\sin A+\sin B+\sin C\) then prove that \(\sin ^{2} A+\sin ^{2} B+\sin ^{2} C\) \(=\cos ^{2} A+\cos ^{2} B+\cos ^{2} C=\frac{3}{2}\)

If \(A=\cos ^{2} \theta+\sin ^{4} \theta\), then find the range of \(A\).

If \(f(x)=\cos [\pi] x+\sin [\pi] x\), where \([,\), is the greatest integer function, then \(f\left(\frac{\pi}{2}\right)\) is (a) 0 (b) \(\cos 3\) (c) \(\cos 4\) (d) None.

Find the maximum or minimum values of: (i) \(3 \cos x+4 \sin x+8\) (ii) \(5 \cos x+3 \cos \left(x+\frac{\pi}{3}\right)+3\)

If \(\frac{\tan 3 A}{\tan A}=k\), show that \(\frac{\sin 3 A}{\sin A}=\frac{2 k}{k-1}\) and \(k\) cannot lie between \(1 / 3\) and 3 .

Let \(f(x)=\left|\begin{array}{ccc}1+\sin ^{2} x & \cos ^{2} x & 4 \sin 2 x \\\ \sin ^{2} x & 1+\cos ^{2} x & 4 \sin 2 x \\ \sin ^{2} x & \cos ^{2} x & 1+4 \sin 2 x\end{array}\right|\) then the maximum value of \(f(x)\) is (a) 0 (b) 2 (c) 6 (d) None.

If \(\cos 25^{\circ}+\sin 25^{\circ}=p\), then find \(\cos 50^{\circ}\).

If \(A+B+C=\pi\), then prove that \(\cot A+\cot B+\cot C-\operatorname{cosec} A \operatorname{cosec} B \operatorname{cosec} C\) \(=\cot A \cdot \cot B \cdot \cot C\)

For any real \(x\), the maximum value of \(\cos ^{2}(\cos x)+\sin ^{2}(\sin x)\) is (a) 1 (b) \(1+\sin ^{2} 1\) (c) \(1+\cos ^{2} 1\) (d) None.

Find the maximum value of \(12 \sin \theta-9 \sin ^{2} \theta\).

If \(\tan \alpha=\frac{1}{\sqrt{x\left(x^{2}+x+1\right)}} \tan \beta=\frac{\sqrt{x}}{\sqrt{\left(x^{2}+x+1\right)}}\) and \(\tan \gamma=\frac{\sqrt{\left(x^{2}+x+1\right)}}{x \sqrt{x}}\) then prove that \(\alpha+\beta=\gamma\)

If \(\operatorname{cosec} \theta-\sin \theta=a^{3}\) and \(\sec \theta-\cos \theta=b^{3}\), then find the value of \(a^{2} b^{2}\left(a^{2}+b^{2}\right)\).

If \(\alpha\) and \(\beta\) are acute angles and \(\cos 2 \alpha=\frac{3 \cos 2 \beta-1}{3-\cos 2 \beta}\), prove that \(\tan \alpha: \tan \beta=\sqrt{2}: 1\).

The minimum value of \(\sin ^{8} x+\cos ^{8} x\) is (a) 0 (b) 1 (c) \(1 / 8\) (d) 2 .

If \(x=\sec \theta-\tan \theta\) and \(y=\operatorname{cosec} \theta+\cot \theta\), then prove that \(x y+x-y+1=0\).

If \(\tan ^{3}\left(\frac{\alpha}{2}+\frac{\pi}{4}\right)=\tan \left(\frac{\beta}{2}+\frac{\pi}{4}\right)\), then prove that \(\sin \beta=\frac{\left(3+\sin ^{2} \alpha\right) \sin \alpha}{1+3 \sin ^{2} \alpha}\)

If \(\frac{\sin ^{4} \theta}{a}+\frac{\cos ^{4} \theta}{b}=\frac{1}{a+b}\), then \(\frac{\sin ^{8} \theta}{a^{3}}+\frac{\cos ^{8} \theta}{b^{3}}\) (a) \(\frac{1}{a^{3}+b^{3}}\) (b) \(\frac{1}{(a+b)^{3}}\) (c) \(\frac{1}{(a-b)^{3}}\) (d) None.

If \(\sin \beta=\frac{1}{5} \sin (2 \alpha+\beta)\), then prove that \(\tan (\alpha+\beta)=\frac{3}{2} \tan \alpha\)

The value of \(\tan \left(\frac{\pi}{7}\right) \tan \left(\frac{2 \pi}{7}\right) \tan \left(\frac{3 \pi}{7}\right)\) is (a) 1 (b) \(\frac{1}{\sqrt{7}}\) (c) \(\sqrt{7}\) (d) None.

If \(\sin x+\sin y=3(\cos x-\cos y)\) then prove that \(\sin (3 x)+\sin (3 y)=0\)

If \(\alpha\) and \(\beta\) are the solutions of \(\sin ^{2} x+a \sin x+b=0\) as well as that of \(\cos ^{2} x+c \cos x+d=0\), then \(\sin (\alpha+\beta)\) is (a) \(\frac{2 b d}{b^{2}+d^{2}}\) (b) \(\frac{a^{2}+c^{2}}{2 a c}\) (c) \(\frac{b^{2}+d^{2}}{2 b d}\) (d) \(\frac{2 a c}{a^{2}+c^{2}}\)

Find the value of \(\tan 70^{\circ}-\tan 20^{\circ}\).

If \(\sec (\varphi-\alpha), \sec \varphi, \sec (\varphi+\alpha)\) are in A.P then prove that \(\cos (\varphi)=\sqrt{2} \cos \left(\frac{\alpha}{2}\right)\)

If \(\sec \theta+\tan \theta=1\), then one of the roots of the equation \(a(b-c) x^{2}+b(c-a) x+c(a-b)=0\) is \(\begin{array}{llll}\text { (a) } \tan \theta & \text { (b) } \sec \theta & \text { (c) } \cos \theta & \text { (d) } \sin \theta \text {. }\end{array}\)