Chapter 1

If \(\tan \left(\frac{x+y}{2}\right), \tan z, \tan \left(\frac{x-y}{2}\right)\) are in G.P then prove that \(\cos (x)=\cos (y) \cos (2 z)\)

If \(\alpha\) is the common +ve root of the equation \(x^{2}-a x+12=0, x^{2}-b x+15=0\) and \(x^{2}-(a+b) x+36=0\) and \(\cos x+\cos 2 x+\cos 3 x=0\), then \(\sin x+\sin 2 x+\sin 3 x\) is (a) 3 (b) \(-3\) (c) 0 (d) 2

Find the value of \(\tan 9^{\circ}-\tan 27^{\circ}-\tan 63^{\circ}+\tan 81^{\circ}\).

Prove that \(\frac{\sec ^{2} \theta-\tan \theta}{\sec ^{2} \theta+\tan \theta}\) lies between \(1 / 3\) and 3 for all real \(\theta\).

For any real \(\theta\), the maximum value of \(\cos ^{2}(\cos \theta)+\sin ^{2}(\sin \theta)\) is (a) 1 (b) \(1+\sin ^{2} 1\) (c) \(1+\cos ^{2} 1\) (d) \(1-\cos ^{2} 1\)

If \(\tan \alpha+\tan \left(\alpha+\frac{\pi}{3}\right)+\tan \left(\alpha+\frac{2 \pi}{3}\right)\) \(=\lambda \tan 3 \alpha\), then find \(\lambda\).

If \(\theta=\frac{\pi}{2^{n}+1}\), then find the value of \(2^{n} \cos (\theta) \cos (2 \theta) \cos \left(2^{2} \theta\right) \ldots \cos \left(2^{n-1} \theta\right)\)

If \(\alpha+\beta=\pi / 2\) and \(\beta+\gamma=\alpha\), then \(\alpha\) is (a) \(2(\tan \beta+\tan \gamma)\) (b) \((\tan \beta+\tan \gamma)\) (c) \((\tan \beta+2 \tan \gamma)\) (d) \((2 \tan \beta+\tan \gamma)\)

Find the value of \(\frac{1}{\sin 10^{\circ}}-\frac{\sqrt{3}}{\cos 10^{\circ}}\).

Find the value of \(\tan \left(6^{\circ}\right) \tan \left(42^{\circ}\right) \tan \left(66^{\circ}\right) \tan \left(78^{\circ}\right)\)

The maximum value of \(\cos \alpha_{1} \cdot \cos \alpha_{2} \cdot \cos \alpha_{3} \ldots \ldots \ldots \ldots . . \cos \alpha_{n}\) under the restriction \(0 \leq \alpha_{1}, \alpha_{2}, \alpha_{3}, \ldots \ldots \ldots \ldots \ldots, \alpha_{n} \leq \frac{\pi}{2}\) and \(\cot \alpha_{1} \cdot \cot \alpha_{2} \cdot \cot \alpha_{3} \ldots \ldots \ldots \ldots \ldots . \cot \alpha_{n}=1\), is (a) \(\frac{1}{2^{n / 2}}\) (b) \(\frac{1}{2^{n}}\) (c) \(\frac{1}{2^{n}}\) (d) 1

If \(2 \cos \theta=x+\frac{1}{x}, 2 \cos \phi=y+\frac{1}{y}\), then find the value of \(\cos (\theta-\varphi)\)

If \(\frac{\tan (\alpha+\beta-\gamma)}{\tan (\alpha-\beta+\gamma)}=\frac{\tan \gamma}{\tan \beta}\), then prove that \(\sin (\beta-\gamma)=0\) or \(\sin 2 \alpha+\sin 2 \beta+\sin 2 \gamma=0\)

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

Prove that \(\frac{\sec 8 \theta-1}{\sec 4 \theta-1}=\tan 8 \theta . \cot 2 \theta\).

If \(A+B+C=\pi\), then prove that \(\cot A+\frac{\sin A}{\sin B \sin C}=\cot B+\frac{\sin B}{\sin A \sin C}\) \(=\cot C+\frac{\sin C}{\sin A \sin B}\)

If \(\tan \beta=2 \sin \alpha \cdot \sin \gamma \cdot \operatorname{cosec}(\alpha+\gamma)\), then \(\cot \alpha, \cot \beta, \cot \gamma\) are in (a) A.P (b) G.P (c) H.P (d) A.G.P

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

The minimum value of the expression \(\sin \alpha+\sin \beta+\sin \gamma\), where \(\alpha, \beta, \gamma\) are real +ve angles satisfying \(\alpha+\beta+\gamma=\pi\), is (a) \(+\mathrm{ve} \quad\) (b) \(-\mathrm{ve}\) (c) 0 (d) \(-3\)

If \(\cos (x-y)=-1\), then prove that \(\cos x+\cos y=0\) and \(\sin x+\sin y=0\)

The value of \(4 \cos 20^{\circ}-\sqrt{3} \cot 20^{\circ}\) is (a) 1 (b) \(-1\) (c) \(-1 / 2\) (d) \(1 / 4\)

If \(\sin A+\sin B=a\) and \(\cos A+\cos B=b\), then find \(\cos (A+B)\)

If \(\sqrt{2} \cos A=\cos B+\cos ^{3} B\) and \(\sqrt{2} \sin A=\sin B\) \(-\sin ^{3} B\) then prove that \(\sin (A-B)=\pm \frac{1}{3}\)

The maximum value of \(4 \sin ^{2} x+3 \cos ^{2} x+\sin \left(\frac{x}{2}\right)+\cos \left(\frac{x}{2}\right.\) (a) \(4+\sqrt{2}\) (b) \(3+\sqrt{2}\) (c) \(4-\sqrt{2}\) (d) 4

Prove that \(\sin \left(9^{\circ}\right)=\frac{1}{4}(\sqrt{3+\sqrt{5}}-\sqrt{5-\sqrt{5}})\)

The value of the expression \(\left(\sqrt{3} \sin 75^{\circ}-\cos 75^{\circ}\right)\) is (a) \(\frac{1}{2}\) (b) \(\frac{1}{\sqrt{2}}\) (c) \(\sqrt{2}\) (d) 2

Prove that, \((2 \cos \theta-1)(2 \cos 2 \theta-1)\left(2 \cos 2^{2} \theta-1\right)\) \(\ldots\left(2 \cos 2^{n-1} \theta-1\right)=\left(\frac{2 \cos \left(2^{n} \theta\right)+1}{2 \cos \theta+1}\right)\)

Find the range of \(f(x)=\sin \left(\sqrt{\frac{\pi^{2}}{36}-x^{2}}\right)\)

The value of \(\left(4+\sec 20^{\circ}\right) \sin 20^{\circ}\) is (a) 1 (b) \(\sqrt{2}\) (c) \(\sqrt{3}\) (d) \(2 \sqrt{3}\)

Prove that, \(\cos \left(9^{\circ}\right)+\sin \left(9^{\circ}\right)=\left(\frac{\sqrt{3+\sqrt{5}}}{2}\right)\)

Find the value of \(\sum_{k=1}^{6}\left(\sin \left(\frac{2 k \pi}{7}\right)-i \cos \left(\frac{2 k \pi}{7}\right)\right)\) where \(i=\sqrt{-1}\).

If \(\left(1+\tan 1^{\circ}\right)\left(1+\tan 2^{\circ}\right) \ldots\left(1+\tan 45^{\circ}\right)=2^{n}\) then the value of \(n\) is (a) 20 (b) 21 (c) 22 (d) 23

If \(\sin x+\cos x=\frac{\sqrt{7}}{2}\), where \(x \in\left[0, \frac{\pi}{4}\right]\) then prove that \(\tan \left(\frac{x}{2}\right)=\left(\frac{\sqrt{7}-2}{3}\right)\)

If \(\frac{\sin x}{\sin y}=\frac{1}{2}\) and \(\frac{\cos x}{\cos y}=\frac{3}{2}\), where \(x, y \in\left(0, \frac{\pi}{2}\right)\), then prove that \(\tan (x+y)=\sqrt{15}\).

If \(\frac{\tan \theta}{\tan \theta-\tan 3 \theta}=\frac{1}{3}\), then find the value of \(\frac{\cot \theta}{\cot (\theta)-\cot (3 \theta)}\)

If \(\sec (x+y)+\sec (x-y)=2 \sec x\), where \(x, y \in\left(0, \frac{\pi}{2}\right)\), then prove that \(\cos x=\sqrt{2} \cos \left(\frac{y}{2}\right)\).

A real root of the equation \(8 x^{3}-6 x-1=0\) is (a) \(\cos \left(\frac{\pi}{5}\right)\) (b) \(\cos \left(\frac{\pi}{9}\right)\) (c) \(\cos \left(\frac{\pi}{18}\right)\) (d) \(\cos \left(\frac{\pi}{36}\right)\)

The value of \(\left(\sqrt{3} \cot \left(20^{\circ}\right)-4 \cos \left(20^{\circ}\right)\right)\) is (a) 1 (b) \(-1\) (c) \(-\frac{\sqrt{3}}{2}\) (d) \(\frac{\sqrt{3}}{2}\)

If \(\tan ^{2}\left(\frac{\pi}{4}+\frac{\theta}{2}\right)=\frac{a}{b}\), then \(\sin (\theta)\) is (a) \(\left(\frac{a-b}{a+b}\right)\) (b) \(-\left(\frac{a-b}{a+b}\right)\) (c) \(\left(\frac{a+b}{a-b}\right)\) (d) \(-\left(\frac{a+b}{a-b}\right)\)

Let \(y=\frac{\sin ^{3} x}{\cos x}-\frac{\cos ^{3} x}{\sin x}, 0

The expression \(\tan \left(55^{\circ}\right) \tan \left(65^{\circ}\right) \tan \left(75^{\circ}\right)\) simplifies to \(\cot \left(x^{\circ}\right), 0

If \(x_{1}\) and \(x_{2}\) are the roots of \(x^{2}+(1-\sin \theta) x-\frac{1}{2} \cos ^{2} \theta=0\), then the maximum value of \(x_{1}^{2}+x_{2}^{2}\) is (a) 2 (b) 3 (c) \(9 / 4\) (d) 4

The value of the expression \(\cos ^{2}\left(\frac{\pi}{8}\right)+\cos ^{2}\left(\frac{3 \pi}{8}\right)+\cos ^{2}\left(\frac{5 \pi}{8}\right)+\cos ^{2}\left(\frac{7 \pi}{8}\right)\) is (a) rational (b) integral (c) prime (d) composite

If \(\tan x=a\), then the value of \(\cot \left(\frac{\pi}{4}-a\right)\) is (a) \(\left(\frac{a-1}{a+1}\right)\) (b) \(\left(\frac{a^{2}-1}{a^{2}+1}\right)\) (c) \(\left(\frac{a^{2}+1}{a^{2}-1}\right)\) (d) \(\left(\frac{a+1}{a-1}\right)\)

If \(\sin \theta+\cos \theta=\frac{1}{5}, 0 \leq \theta \leq \pi\), then \(\tan \theta\) (a) \(3 / 4\) (b) \(4 / 3\) (c) \(-3 / 4\) (d) \(-4 / 3\)