Class 11 Mathematics Trigonometric Functions MCQs Set 08

Trigonometrical Ratios of Sum & Difference of Two Angles, Transformation of Product into Sum & Difference, Transformation of Sum & Difference into Product

Question. If A, B, C, D are the angles of a cyclic quadrilateral then \( \cos A + \cos B + \cos C + \cos D = \)
(a) \( 2(\cos A + \cos C) \)
(b) \( 2(\cos A + \cos B) \)
(c) \( 2(\cos A + \cos D) \)
(d) 0
Answer: (d)

Question. \( \frac{\cos 17^\circ + \sin 17^\circ}{\cos 17^\circ - \sin 17^\circ} = \)
(a) \( \tan 62^\circ \)
(b) \( \tan 56^\circ \)
(c) \( \tan 54^\circ \)
(d) \( \tan 73^\circ \)
Answer: (a)

Question. \( \cot(45^\circ + \theta) \cot(45^\circ - \theta) = \)
(a) -1
(b) 0
(c) 1
(d) \( \infty \)
Answer: (c)

Question. \( \tan 75^\circ - \cot 75^\circ = \)
(a) \( 2\sqrt{3} \)
(b) \( 2 + \sqrt{3} \)
(c) \( 2 - \sqrt{3} \)
(d) None of these
Answer: (a)

Question. \( \sqrt{3} \csc 20^\circ - \sec 20^\circ = \)
(a) 2
(b) \( \frac{2 \sin 20^\circ}{\sin 40^\circ} \)
(c) 4
(d) \( \frac{4 \sin 20^\circ}{\sin 40^\circ} \)
Answer: (c)

Question. \( \sin 15^\circ + \cos 105^\circ = \)
(a) 0
(b) \( 2 \sin 15^\circ \)
(c) \( \cos 15^\circ + \sin 15^\circ \)
(d) \( \sin 15^\circ - \cos 15^\circ \)
Answer: (a)

Question. If \( \cos(A + B) = \alpha \cos A \cos B + \beta \sin A \sin B \), then \( (\alpha, \beta) = \)
(a) \( (-1, -1) \)
(b) \( (-1, 1) \)
(c) \( (1, -1) \)
(d) \( (1, 1) \)
Answer: (c)

Question. \( \cos^2 \alpha + \cos^2(\alpha + 120^\circ) + \cos^2(\alpha - 120^\circ) \) is equal to
(a) 3/2
(b) 1
(c) 1/2
(d) 0
Answer: (a)

Question. The value of \( \cos 105^\circ + \sin 105^\circ \) is
(a) 1/2
(b) 1
(c) \( \sqrt{2} \)
(d) \( \frac{1}{\sqrt{2}} \)
Answer: (d)

Question. \( \cos^2 48^\circ - \sin^2 12^\circ = \)
(a) \( \frac{\sqrt{5} - 1}{4} \)
(b) \( \frac{\sqrt{5} + 1}{8} \)
(c) \( \frac{\sqrt{3} - 1}{4} \)
(d) \( \frac{\sqrt{3} + 1}{2\sqrt{2}} \)
Answer: (b)

Question. \( \sin 20^\circ \sin 40^\circ \sin 60^\circ \sin 80^\circ = \)
(a) -3/16
(b) 5/16
(c) 3/16
(d) -5/16
Answer: (c)

Question. \( \cos 20^\circ \cos 40^\circ \cos 80^\circ = \)
(a) 1/2
(b) 1/4
(c) 1/6
(d) 1/8
Answer: (d)

Question. \( \cos \frac{2\pi}{15} \cos \frac{4\pi}{15} \cos \frac{8\pi}{15} \cos \frac{16\pi}{15} = \)
(a) 1/2
(b) 1/4
(c) 1/8
(d) 1/16
Answer: (d)

Question. If \( x = \cos 10^\circ \cos 20^\circ \cos 40^\circ \), then the value of x is
(a) \( \frac{1}{4} \tan 10^\circ \)
(b) \( \frac{1}{8} \cot 10^\circ \)
(c) \( \frac{1}{8} \csc 10^\circ \)
(d) \( \frac{1}{8} \sec 10^\circ \)
Answer: (b)

Question. The value of \( \cos 52^\circ + \cos 68^\circ + \cos 172^\circ \) is
(a) 0
(b) 1
(c) 2
(d) 3/2
Answer: (a)

Question. \( \cos 40^\circ + \cos 80^\circ + \cos 160^\circ + \cos 240^\circ = \)
(a) 0
(b) 1
(c) 1/2
(d) -1/2
Answer: (d)

Question. \( 1 + \cos 56^\circ + \cos 58^\circ - \cos 66^\circ = \)
(a) \( 2 \cos 28^\circ \cos 29^\circ \cos 33^\circ \)
(b) \( 4 \cos 28^\circ \cos 29^\circ \cos 33^\circ \)
(c) \( 4 \cos 28^\circ \cos 29^\circ \sin 33^\circ \)
(d) \( 2 \cos 28^\circ \cos 29^\circ \sin 33^\circ \)
Answer: (b)

Question. \( \tan 5x \tan 3x \tan 2x = \)
(a) \( \tan 5x - \tan 3x - \tan 2x \)
(b) \( \frac{\sin 5x - \sin 3x - \sin 2x}{\cos 5x - \cos 3x - \cos 2x} \)
(c) 0
(d) None of these
Answer: (a)

Question. If \( \cos \alpha + \cos \beta = 0 = \sin \alpha + \sin \beta \), then \( \cos 2\alpha + \cos 2\beta = \)
(a) \( -2 \sin(\alpha + \beta) \)
(b) \( -2 \cos(\alpha + \beta) \)
(c) \( 2 \sin(\alpha + \beta) \)
(d) \( 2 \cos(\alpha + \beta) \)
Answer: (b)

Question. If \( \tan A = -\frac{1}{2} \) and \( \tan B = -\frac{1}{3} \), then \( A + B = \)
(a) \( \frac{\pi}{4} \)
(b) \( \frac{3\pi}{4} \)
(c) \( \frac{5\pi}{4} \)
(d) None of these
Answer: (b)

Question. If \( \cos(A - B) = \frac{3}{5} \) and \( \tan A \tan B = 2 \), then
(a) \( \cos A \cos B = \frac{1}{5} \)
(b) \( \sin A \sin B = -\frac{2}{5} \)
(c) \( \cos A \cos B = -\frac{1}{5} \)
(d) \( \sin A \sin B = -\frac{1}{5} \)
Answer: (a)

Question. \( \frac{\sin 3\theta - \cos 3\theta}{\sin \theta + \cos \theta} + 1 = \)
(a) \( 2 \sin 2\theta \)
(b) \( 2 \cos 2\theta \)
(c) \( \tan 2\theta \)
(d) \( \cot 2\theta \)
Answer: (a)

Question. \( \tan 3A - \tan 2A - \tan A = \)
(a) \( \tan 3A \tan 2A \tan A \)
(b) \( -\tan 3A \tan 2A \tan A \)
(c) \( \tan A \tan 2A - \tan 2A \tan 3A - \tan 3A \tan A \)
(d) None of these
Answer: (a)

Question. If \( \cos A = m \cos B \), then
(a) \( \cot \frac{A+B}{2} = \frac{m+1}{m-1} \tan \frac{B-A}{2} \)
(b) \( \tan \frac{A+B}{2} = \frac{m+1}{m-1} \cot \frac{B-A}{2} \)
(c) \( \cot \frac{A+B}{2} = \frac{m+1}{m-1} \tan \frac{A-B}{2} \)
(d) None of these
Answer: (a)

Question. The value of \( \cos 12^\circ + \cos 84^\circ + \cos 156^\circ + \cos 132^\circ \) is
(a) 1/2
(b) 1
(c) -1/2
(d) 1/8
Answer: (c)

Question. \( \tan 100^\circ + \tan 125^\circ + \tan 100^\circ \tan 125^\circ = \)
(a) 0
(b) \( \frac{1}{2} \)
(c) -1
(d) 1
Answer: (d)

Question. If \( \cos P = \frac{1}{7} \) and \( \cos Q = \frac{13}{14} \) where P and Q both are acute angles. Then the value of \( P - Q \) is
(a) \( 30^\circ \)
(b) \( 60^\circ \)
(c) \( 45^\circ \)
(d) \( 75^\circ \)
Answer: (b)

Question. If \( \sin A = \frac{1}{\sqrt{10}} \) and \( \sin B = \frac{1}{\sqrt{5}} \), where A and B are positive acute angles, then \( A + B = \)
(a) \( \pi \)
(b) \( \frac{\pi}{2} \)
(c) \( \frac{\pi}{3} \)
(d) \( \frac{\pi}{4} \)
Answer: (d)

Question. \( \sin \left(\frac{\pi}{10}\right) \sin \left(\frac{3\pi}{10}\right) = \)
(a) 1/2
(b) -1/2
(c) 1/4
(d) 1
Answer: (c)

Question. \( \sin 50^\circ - \sin 70^\circ + \sin 10^\circ = \)
(a) 1
(b) 0
(c) 1/2
(d) 2
Answer: (b)

Question. If \( \sin A = \sin B \) and \( \cos A = \cos B \), then
(a) \( \sin \frac{A-B}{2} = 0 \)
(b) \( \sin \frac{A+B}{2} = 0 \)
(c) \( \cos \frac{A-B}{2} = 0 \)
(d) \( \cos(A+B) = 0 \)
Answer: (a)

Question. \( \sin 12^\circ \sin 48^\circ \sin 54^\circ = \)
(a) 1/16
(b) 1/32
(c) 1/8
(d) 1/4
Answer: (c)

Question. If \( (1 + \tan \theta)(1 + \tan \phi) = 2 \), then \( \theta + \phi = \)
(a) \( 30^\circ \)
(b) \( 45^\circ \)
(c) \( 60^\circ \)
(d) \( 75^\circ \)
Answer: (b)

Question. \( \cos^2\left(\frac{\pi}{6}+\theta\right) - \sin^2\left(\frac{\pi}{6}-\theta\right) = \)
(a) \( \frac{1}{2} \cos 2\theta \)
(b) 0
(c) \( -\frac{1}{2} \cos 2\theta \)
(d) \( \frac{1}{2} \)
Answer: (a)

Question. If \( \sin \theta + \sin 2\theta + \sin 3\theta = \sin \alpha \) and \( \cos \theta + \cos 2\theta + \cos 3\theta = \cos \alpha \), then \( \theta \) is equal to
(a) \( \alpha/2 \)
(b) \( \alpha \)
(c) \( 2\alpha \)
(d) \( \alpha/6 \)
Answer: (a)

Question. \( \cos \alpha \sin(\beta - \gamma) + \cos \beta \sin(\gamma - \alpha) + \cos \gamma \sin(\alpha - \beta) = \)
(a) 0
(b) 1/2
(c) 1
(d) \( 4 \cos \alpha \cos \beta \cos \gamma \)
Answer: (a)

Question. Given that \( \cos \left(\frac{\alpha-\beta}{2}\right) = 2 \cos \left(\frac{\alpha+\beta}{2}\right) \), then \( \tan \frac{\alpha}{2} \tan \frac{\beta}{2} \) is equal to
(a) \( \frac{1}{2} \)
(b) \( \frac{1}{3} \)
(c) \( \frac{1}{4} \)
(d) \( \frac{1}{8} \)
Answer: (b)

Question. If \( \sin A + \sin B = C, \cos A + \cos B = D \), then the value of \( \sin(A+B) = \)
(a) \( CD \)
(b) \( \frac{CD}{C^2 + D^2} \)
(c) \( \frac{C^2 + D^2}{2CD} \)
(d) \( \frac{2CD}{C^2 + D^2} \)
Answer: (d)

Question. If \( A + B = 225^\circ \), then \( \frac{\cot A}{1 + \cot A} \cdot \frac{\cot B}{1 + \cot B} = \)
(a) 1
(b) -1
(c) 0
(d) \( \frac{1}{2} \)
Answer: (d)

Question. \( \frac{1}{\sin 10^\circ} - \frac{\sqrt{3}}{\cos 10^\circ} \)
(a) 0
(b) 1
(c) 2
(d) 4
Answer: (d)

Question. \( \frac{\sin 3\theta + \sin 5\theta + \sin 7\theta + \sin 9\theta}{\cos 3\theta + \cos 5\theta + \cos 7\theta + \cos 9\theta} = \)
(a) \( \tan 3\theta \)
(b) \( \cot 3\theta \)
(c) \( \tan 6\theta \)
(d) \( \cot 6\theta \)
Answer: (c)

Question. If \( \cos(\theta - \alpha), \cos \theta \) and \( \cos(\theta + \alpha) \) are in H.P., then \( \cos \theta \sec \frac{\alpha}{2} \) is equal to
(a) \( \pm \sqrt{2} \)
(b) \( \pm \sqrt{3} \)
(c) \( \pm 1/\sqrt{2} \)
(d) None of these
Answer: (a)

Question. \( \frac{\sin(B + A) + \cos(B - A)}{\sin(B - A) + \cos(B + A)} = \)
(a) \( \frac{\cos B + \sin B}{\cos B - \sin B} \)
(b) \( \frac{\cos A + \sin A}{\cos A - \sin A} \)
(c) \( \frac{\cos A - \sin A}{\cos A + \sin A} \)
(d) None of these
Answer: (b)

Question. If \( \sin 2x = n \sin 2y \), then the value of \( \frac{\tan(x+y)}{\tan(x-y)} \) is
(a) \( \frac{n+1}{n-1} \)
(b) \( \frac{n-1}{n+1} \)
(c) \( \frac{1-n}{n+1} \)
(d) \( \frac{1+n}{1-n} \)
Answer: (a)

Question. If \( 3 \sin \alpha = 5 \sin \beta \), then \( \frac{\tan \frac{\alpha + \beta}{2}}{\tan \frac{\alpha - \beta}{2}} = \)
(a) 1
(b) 2
(c) 3
(d) 4
Answer: (d)

Question. If \( \frac{\pi}{2} < \alpha < \pi, \pi < \beta < \frac{3\pi}{2}, \sin \alpha = \frac{15}{17} \) and \( \tan \beta = \frac{12}{5} \), the value of \( \sin(\beta - \alpha) \) is
(a) \( \frac{-171}{221} \)
(b) \( \frac{-21}{221} \)
(c) \( \frac{21}{221} \)
(d) \( \frac{17}{221} \)
Answer: (a)

Question. \( \cos^2 76^\circ + \cos^2 16^\circ - \cos 76^\circ \cos 16^\circ = \)
(a) \( -\frac{1}{4} \)
(b) \( \frac{1}{2} \)
(c) 0
(d) \( \frac{3}{4} \)
Answer: (d)

Question. The value of \( \cos^2 \frac{\pi}{12} + \cos^2 \frac{\pi}{4} + \cos^2 \frac{5\pi}{12} \) is
(a) \( \frac{3}{2} \)
(b) \( \frac{2}{3} \)
(c) \( \frac{3 + \sqrt{3}}{2} \)
(d) \( \frac{2}{3 + \sqrt{3}} \)
Answer: (a)

Question. If angle \( \theta \) be divided into two parts such that the tangents of one part is K times the tangent of the other and \( \phi \) is their difference, then \( \sin \theta = \)
(a) \( \frac{K+1}{K-1} \sin \phi \)
(b) \( \frac{K-1}{K+1} \sin \phi \)
(c) \( \frac{2K-1}{2K+1} \sin \phi \)
(d) None of these
Answer: (a)

Question. If \( \tan \alpha, \tan \beta \) are the roots of the equation \( x^2 + px + q = 0 (p \neq 0) \), then
(a) \( \sin^2(\alpha + \beta) + p \sin(\alpha + \beta) \cos(\alpha + \beta) + q \cos^2(\alpha + \beta) = q \)
(b) \( \tan(\alpha + \beta) = \frac{p}{q-1} \)
(c) \( \cos(\alpha + \beta) = 1 - q \)
(d) \( \sin(\alpha + \beta) = -p \)
Answer: (b)

Question. If \( \tan \alpha \) equals the integral solution of the inequality \( 4x^2 - 16x + 15 < 0 \) and \( \cos \beta \) equals to the slope of the bisector of first quadrant, then \( \sin(\alpha + \beta) \sin(\alpha - \beta) \) is equal to
(a) \( \frac{3}{5} \)
(b) \( -\frac{3}{5} \)
(c) \( \frac{2}{\sqrt{5}} \)
(d) \( \frac{4}{5} \)
Answer: (d)