JEE Mathematics Trigonometrical Functions and Identities MCQs Set A

Choose the most appropriate option (a, b, c or d).

Question. If \( \tan \theta = a - \frac{1}{4a} \), then \( \sec \theta - \tan \theta \) is equal to
(a) \( -2a, \frac{1}{2a} \)
(b) \( -\frac{1}{2a}, 2a \)
(c) \( 2a \)
(d) \( \frac{1}{2a}, 2a \)
Answer: (a) \( -2a, \frac{1}{2a} \)

Question. \( \sec^2 \theta = \frac{4xy}{(x + y)^2} \), where \( x \in \mathbb{R}, y \in \mathbb{R} \), is true if and only if
(a) \( x + y \neq 0 \)
(b) \( x = y, x \neq 0 \)
(c) \( x = y \)
(d) \( x \neq 0, y \neq 0 \)
Answer: (b) \( x = y, x \neq 0 \)

Question. \( \sin^2 \theta = \frac{(x + y)^2}{4xy} \), where \( x \in \mathbb{R} \), gives real \( \theta \) if and only if
(a) \( x + y = 0 \)
(b) \( x = y \)
(c) \( |x| = |y| \neq 0 \)
(d) None of the options
Answer: (c) \( |x| = |y| \neq 0 \)

Question. \( \csc \theta = \frac{x^2 - y^2}{x^2 + y^2} \), where \( x \in \mathbb{R}, y \in \mathbb{R} \), gives real \( \theta \) if and only if
(a) \( x = y \neq 0 \)
(b) \( |x| = |y| \neq 0 \)
(c) \( x + y = 0, x \neq 0 \)
(d) None of the options
Answer: (d) None of the options

Question. If \( \sin \theta + \csc \theta = 2 \) then the value of \( \sin^8 \theta + \csc^8 \theta \) is equal to
(a) 2
(b) \( 2^8 \)
(c) \( 2^4 \)
(d) None of the options
Answer: (a) 2

Question. If \( x = r \sin \theta \cos \phi, y = r \sin \theta \sin \phi \) and \( z = r \cos \theta \), then the value of \( x^2 + y^2 + z^2 \) is independent of
(a) \( \theta, \phi \)
(b) \( r, \theta \)
(c) \( r, \phi \)
(d) \( r \)
Answer: (a) \( \theta, \phi \)

Question. Let \( p = a \cos \theta - b \sin \theta \). Then for all real \( \theta \)
(a) \( p > \sqrt{a^2 + b^2} \)
(b) \( p < -\sqrt{a^2 + b^2} \)
(c) \( -\sqrt{a^2 + b^2} \leq p \leq \sqrt{a^2 + b^2} \)
(d) None of the options
Answer: (c) \( -\sqrt{a^2 + b^2} \leq p \leq \sqrt{a^2 + b^2} \)

Question. If \( 0^\circ < \theta < 180^\circ \) then \( \sqrt{2 + \sqrt{2 + \sqrt{2 + \dots + \sqrt{2(1 + \cos \theta)}}}} \), there being \( n \) number of 2's, is equal to
(a) \( 2 \cos \frac{\theta}{2^n} \)
(b) \( 2 \cos \frac{\theta}{2^{n-1}} \)
(c) \( 2 \cos \frac{\theta}{2^{n+1}} \)
(d) None of the options
Answer: (a) \( 2 \cos \frac{\theta}{2^n} \)

Question. The value of \( \tan \frac{\pi}{16} + 2 \tan \frac{\pi}{8} + 4 \) is equal to
(a) \( \cot \frac{\pi}{8} \)
(b) \( \cot \frac{\pi}{16} \)
(c) \( \cot \frac{\pi}{16} - 4 \)
(d) None of the options
Answer: (b) \( \cot \frac{\pi}{16} \)

Question. The value of \( \sin 78^\circ - \sin 66^\circ - \sin 42^\circ + \sin 6^\circ \) is
(a) \( \frac{1}{2} \)
(b) \( -\frac{1}{2} \)
(c) \( -1 \)
(d) None of the options
Answer: (b) \( -\frac{1}{2} \)

Question. The value of \( \sqrt{3} \csc 20^\circ - \sec 20^\circ \) is equal to
(a) 2
(b) 4
(c) \( 2 \cdot \frac{\sin 20^\circ}{\sin 40^\circ} \)
(d) \( 4 \cdot \frac{\sin 20^\circ}{\sin 40^\circ} \)
Answer: (b) 4

Question. The maximum value of \( 1 + \sin\left(\frac{\pi}{4} + \theta\right) + 2 \cos\left(\frac{\pi}{4} - \theta\right) \) for real values of \( \theta \) is
(a) 3
(b) 5
(c) 4
(d) None of the options
Answer: (c) 4

Question. The minimum value of \( \cos 2\theta + \cos \theta \) of real values of \( \theta \) is
(a) \( -\frac{9}{8} \)
(b) 0
(c) -2
(d) None of the options
Answer: (a) \( -\frac{9}{8} \)

Question. The value of \( \csc 10^\circ - \sqrt{3} \sec 10^\circ \) is equal to
(a) \( \frac{1}{2} \)
(b) 2
(c) 4
(d) 8
Answer: (c) 4

Question. The least value of \( \cos^2 \theta - 6 \sin \theta \cos \theta + 3 \sin^2 \theta + 2 \) is
(a) \( 4 + \sqrt{10} \)
(b) \( 4 - \sqrt{10} \)
(c) 0
(d) None of the options
Answer: (b) \( 4 - \sqrt{10} \)

Question. If \( \cos^4 \theta \sec^2 \alpha \), \( \frac{1}{2} \) and \( \sin^4 \theta \csc^2 \alpha \) are in AP then \( \cos^8 \theta \cos^6 \alpha \), \( \frac{1}{2} \) and \( \sin^8 \theta \csc^6 \alpha \) are in
(a) AP
(b) GP
(c) HP
(d) None of the options
Answer: (a) AP

Question. If \( \tan \frac{\pi}{9} \), \( x \) and \( \tan \frac{5\pi}{18} \) are in AP and \( \tan \frac{\pi}{9} \), \( y \) and \( \tan \frac{7\pi}{18} \) are also in AP then
(a) \( 2x = y \)
(b) \( x > y \)
(c) \( x = y \)
(d) None of the options
Answer: (a) \( 2x = y \)

Question. If \( \cos(x - y) \), \( \cos x \) and \( \cos(x + y) \) are in HP then \( \cos x \sec \frac{y}{2} \) equals
(a) 1
(b) 2
(c) \( \sqrt{2} \)
(d) None of the options
Answer: (c) \( \sqrt{2} \)

Question. If \( 2 \sin \alpha \cos \beta \sin \gamma = \sin \beta \sin(\alpha + \gamma) \) then \( \tan \alpha, \tan \beta \) and \( \tan \gamma \) are in
(a) AP
(b) GP
(c) HP
(d) None of the options
Answer: (c) HP

Question. If \( \tan \alpha = \sqrt{a} \), where \( a \) is a rational number which is not a perfect square, then which of the following is a rational number?
(a) \( \sin 2\alpha \)
(b) \( \tan 2\alpha \)
(c) \( \cos 2\alpha \)
(d) None of the options
Answer: (c) \( \cos 2\alpha \)

Question. Let \( f(\theta) = \frac{\cot \theta}{1 + \cot \theta} \) and \( \alpha + \beta = \frac{5\pi}{4} \). Then the value of \( f(\alpha) \cdot f(\beta) \) is
(a) 2
(b) \( -\frac{1}{2} \)
(c) \( \frac{1}{2} \)
(d) None of the options
Answer: (c) \( \frac{1}{2} \)

Question. If \( \tan \frac{\alpha}{2} \) and \( \tan \frac{\beta}{2} \) are the roots of the equation \( 8x^2 - 26x + 15 = 0 \), then \( \cos(\alpha + \beta) \) is equal to
(a) \( -\frac{627}{725} \)
(b) \( \frac{627}{725} \)
(c) -1
(d) None of the options
Answer: (a) \( -\frac{627}{725} \)

Question. If \( \sin \alpha + \sin \beta = a \) and \( \cos \alpha - \cos \beta = b \), then \( \tan \frac{\alpha - \beta}{2} \) is equal to
(a) \( -\frac{a}{b} \)
(b) \( -\frac{b}{a} \)
(c) \( \sqrt{a^2 + b^2} \)
(d) None of the options
Answer: (b) \( -\frac{b}{a} \)

Question. If \( 0 < \beta < \alpha < \frac{\pi}{4} \), \( \cos(\alpha + \beta) = \frac{3}{5} \) and \( \sin(\alpha - \beta) = \frac{4}{5} \), then \( \sin 2\alpha \) is equal to
(a) 1
(b) 0
(c) 2
(d) None of the options
Answer: (a) 1

Question. If \( \cos \alpha = \frac{1}{2}\left(x + \frac{1}{x}\right) \) and \( \cos \beta = \frac{1}{2}\left(y + \frac{1}{y}\right) \), then \( \cos(\alpha - \beta) \) is equal to
(a) \( \frac{x}{y} + \frac{y}{x} \)
(b) \( xy + \frac{1}{xy} \)
(c) \( \frac{1}{2}\left(\frac{x}{y} + \frac{y}{x}\right) \)
(d) None of the options
Answer: (c) \( \frac{1}{2}\left(\frac{x}{y} + \frac{y}{x}\right) \)

Question. If \( \frac{2 \sin \alpha}{1 + \sin \alpha + \cos \alpha} = \lambda \), then \( \frac{1 + \sin \alpha - \cos \alpha}{1 + \sin \alpha} \) is equal to
(a) \( \frac{1}{\lambda} \)
(b) \( \lambda \)
(c) \( 1 - \lambda \)
(d) \( 1 + \lambda \)
Answer: (b) \( \lambda \)

Question. If \( |\tan A| < 1 \), and \( |A| \) is acute then \( \frac{\sqrt{1 + \sin 2A} + \sqrt{1 - \sin 2A}}{\sqrt{1 + \sin 2A} - \sqrt{1 - \sin 2A}} \) is equal to
(a) \( \tan A \)
(b) \( -\tan 3\theta \)
(c) \( \cot A \)
(d) \( -\cot A \)
Answer: (c) \( \cot A \)

Question. \( \tan \theta \cdot \tan\left(\frac{\pi}{3} + \theta\right) \cdot \tan\left(\frac{\pi}{3} - \theta\right) \) is equal to
(a) \( \tan 2\theta \)
(b) \( \tan 3\theta \)
(c) \( \tan^3 \theta \)
(d) None of the options
Answer: (b) \( \tan 3\theta \)

Question. The set of all possible values of \( \alpha \) in \( [-\pi, \pi] \) such that \( \sqrt{\frac{1 - \sin \alpha}{1 + \sin \alpha}} \) is equal to \( \sec \alpha - \tan \alpha \) is
(a) \( [0, \pi/2) \)
(b) \( [0, \pi/2) \cup (\pi/2, \pi] \)
(c) \( [-\pi, 0] \)
(d) \( (-\pi/2, \pi/2) \)
Answer: (d) \( (-\pi/2, \pi/2) \)

Question. For all real values of \( \theta \), \( \cot \theta - 2 \cot 2\theta \) is equal to
(a) \( \tan 2\theta \)
(b) \( \tan \theta \)
(c) \( -\cot 3\theta \)
(d) None of the options
Answer: (b) \( \tan \theta \)

Question. Let \( a = \cos A + \cos B - \cos(A + B) \) and \( b = 4 \sin \frac{A}{2} \sin \frac{B}{2} \cos \frac{A + B}{2} \). Then \( a - b \) is equal to
(a) 1
(b) 0
(c) -1
(d) None of the options
Answer: (a) 1

Question. If \( \tan \theta + \tan\left(\theta + \frac{\pi}{3}\right) + \tan\left(\theta - \frac{\pi}{3}\right) = k \tan 3\theta \), then \( k \) is equal to
(a) 1
(b) 3
(c) \( \frac{1}{3} \)
(d) None of the options
Answer: (b) 3

Question. If \( a \sec \alpha - c \tan \alpha = d \) and \( b \sec \alpha + d \tan \alpha = c \) then
(a) \( a^2 + c^2 = b^2 + d^2 \)
(b) \( a^2 + d^2 = b^2 + c^2 \)
(c) \( a^2 + b^2 = c^2 + d^2 \)
(d) \( ab = cd \)
Answer: (c) \( a^2 + b^2 = c^2 + d^2 \)

Question. If \( \cos 20^\circ - \sin 20^\circ = p \) then \( \cos 40^\circ \) is equal to
(a) \( -p\sqrt{2 - p^2} \)
(b) \( p\sqrt{2 - p^2} \)
(c) \( p + \sqrt{2 - p^2} \)
(d) None of the options
Answer: (b) \( p\sqrt{2 - p^2} \)

Question. If \( 3 \sin \theta + 4 \cos \theta = 5 \) then the value of \( 4 \sin \theta - 3 \cos \theta \) is
(a) 0
(b) 5
(c) 1
(d) None of the options
Answer: (a) 0

Choose the correct options. One or more options may be correct

Question. If \( x = \sin(\alpha - \beta) \cdot \sin(\gamma - \delta) \), \( y = \sin(\beta - \gamma) \cdot \sin(\alpha - \delta) \) and \( z = \sin(\gamma - \alpha) \cdot \sin(\beta - \delta) \) then
(a) \( x + y + z = 0 \)
(b) \( x + y - z = 0 \)
(c) \( y + z - x = 0 \)
(d) \( x^3 + y^3 + z^3 = 3xyz \)
Answer: (a) \( x + y + z = 0 \)
(d) \( x^3 + y^3 + z^3 = 3xyz \)

Question. \( \sin \frac{15\pi}{32} \cdot \sin \frac{7\pi}{16} \cdot \sin \frac{3\pi}{8} \) is equal to
(a) \( \frac{1}{8\sqrt{2} \cos \frac{15\pi}{32}} \)
(b) \( \frac{1}{8 \sin \frac{\pi}{32}} \)
(c) \( \frac{1}{4\sqrt{2}} \csc \frac{\pi}{16} \)
(d) \( \frac{1}{8\sqrt{2}} \csc \frac{\pi}{32} \)
Answer: (a) \( \frac{1}{8\sqrt{2} \cos \frac{15\pi}{32}} \)
(d) \( \frac{1}{8\sqrt{2}} \csc \frac{\pi}{32} \)

Question. Which of the following is a rational number?
(a) \( \sin 15^\circ \)
(b) \( \cos 15^\circ \)
(c) \( \sin 15^\circ \)
(d) \( \sin 15^\circ \cdot \cos 75^\circ \)
Answer: (c) \( \sin 15^\circ \)

Question. If \( a = \frac{1}{5 \cos x + 12 \sin x} \) then for all real \( x \)
(a) the least positive value of \( a \) is \( \frac{1}{13} \)
(b) the greatest negative value of \( a \) is \( -\frac{1}{13} \)
(c) \( a \leq \frac{1}{13} \)
(d) \( -\frac{1}{13} \leq a \leq \frac{1}{13} \)
Answer: (a) the least positive value of \( a \) is \( \frac{1}{13} \)
(b) the greatest negative value of \( a \) is \( -\frac{1}{13} \)

Question. Let \( y = \sin^2 x + \cos^4 x \). Then for all real \( x \)
(a) the maximum value of \( y \) is 2
(b) the minimum value of \( y \) is \( \frac{3}{4} \)
(c) \( y \leq \frac{1}{4} \)
(d) \( y \geq -1 \)
Answer: (b) the minimum value of \( y \) is \( \frac{3}{4} \)
(c) \( y \leq \frac{1}{4} \)

Question. Let \( y = \sin x \cdot \sin(60^\circ + x) \cdot \sin(60^\circ - x) \). Then for all real \( x \)
(a) the minimum value of \( y \) is \( -\frac{1}{4} \)
(b) the maximum value of \( y \) is 1
(c) \( y \leq \frac{1}{4} \)
(d) \( y \geq -1 \)
Answer: (a) the minimum value of \( y \) is \( -\frac{1}{4} \)
(c) \( y \leq \frac{1}{4} \)

Question. Let \( f_n(\theta) = \tan \frac{\theta}{2} (1 + \sec \theta)(1 + \sec 2\theta)(1 + \sec 4\theta) \dots (1 + \sec 2^n \theta) \). Then
(a) \( f_2\left(\frac{\pi}{16}\right) = 1 \)
(b) \( f_3\left(\frac{\pi}{32}\right) = 1 \)
(c) \( f_4\left(\frac{\pi}{64}\right) = 1 \)
(d) \( f_5\left(\frac{\pi}{128}\right) = 1 \)
Answer: (a) \( f_2\left(\frac{\pi}{16}\right) = 1 \)
(b) \( f_3\left(\frac{\pi}{32}\right) = 1 \)
(c) \( f_4\left(\frac{\pi}{64}\right) = 1 \)
(d) \( f_5\left(\frac{\pi}{128}\right) = 1 \)

Question. Let \( 0 \leq \theta < \frac{\pi}{2} \) and \( x = X \cos \theta + Y \sin \theta, y = X \sin \theta - Y \cos \theta \) such that \( x^2 + 4xy + y^2 = aX^2 + bY^2 \), where \( a, b \) are constants. Then
(a) \( a = -1, b = 3 \)
(b) \( \theta = \frac{\pi}{4} \)
(c) \( a = 3, b = -1 \)
(d) \( \theta = \frac{\pi}{3} \)
Answer: (b) \( \theta = \frac{\pi}{4} \)
(c) \( a = 3, b = -1 \)