Trigonometric Ratio and Identites · Mathematics · JEE Main

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MCQ (Single Correct Answer)

1

If for $\theta \in\left[-\frac{\pi}{3}, 0\right]$, the points $(x, y)=\left(3 \tan \left(\theta+\frac{\pi}{3}\right), 2 \tan \left(\theta+\frac{\pi}{6}\right)\right)$ lie on $x y+\alpha x+\beta y+\gamma=0$, then $\alpha^2+\beta^2+\gamma^2$ is equal to :

JEE Main 2025 (Online) 7th April Morning Shift
2

If $10 \sin ^4 \theta+15 \cos ^4 \theta=6$, then the value of $\frac{27 \operatorname{cosec}^6 \theta+8 \sec ^6 \theta}{16 \sec ^8 \theta}$ is

JEE Main 2025 (Online) 4th April Morning Shift
3

If $\sin x + \sin^2 x = 1$, $x \in \left(0, \frac{\pi}{2}\right)$, then

$(\cos^{12} x + \tan^{12} x) + 3(\cos^{10} x + \tan^{10} x + \cos^8 x + \tan^8 x) + (\cos^6 x + \tan^6 x)$ is equal to:

JEE Main 2025 (Online) 29th January Evening Shift
4
If $\sum\limits_{r=1}^{13}\left\{\frac{1}{\sin \left(\frac{\pi}{4}+(r-1) \frac{\pi}{6}\right) \sin \left(\frac{\pi}{4}+\frac{r \pi}{6}\right)}\right\}=a \sqrt{3}+b, a, b \in Z$, then $a^2+b^2$ is equal to :
JEE Main 2025 (Online) 28th January Evening Shift
5

Let the range of the function $f(x)=6+16 \cos x \cdot \cos \left(\frac{\pi}{3}-x\right) \cdot \cos \left(\frac{\pi}{3}+x\right) \cdot \sin 3 x \cdot \cos 6 x, x \in \mathbf{R}$ be $[\alpha, \beta]$. Then the distance of the point $(\alpha, \beta)$ from the line $3 x+4 y+12=0$ is :

JEE Main 2025 (Online) 23rd January Evening Shift
6

The value of $\left(\sin 70^{\circ}\right)\left(\cot 10^{\circ} \cot 70^{\circ}-1\right)$ is

JEE Main 2025 (Online) 23rd January Morning Shift
7

If the value of $$\frac{3 \cos 36^{\circ}+5 \sin 18^{\circ}}{5 \cos 36^{\circ}-3 \sin 18^{\circ}}$$ is $$\frac{a \sqrt{5}-b}{c}$$, where $$a, b, c$$ are natural numbers and $$\operatorname{gcd}(a, c)=1$$, then $$a+b+c$$ is equal to :

JEE Main 2024 (Online) 8th April Evening Shift
8

If $$\sin x=-\frac{3}{5}$$, where $$\pi< x <\frac{3 \pi}{2}$$, then $$80\left(\tan ^2 x-\cos x\right)$$ is equal to

JEE Main 2024 (Online) 8th April Morning Shift
9

Suppose $$\theta \in\left[0, \frac{\pi}{4}\right]$$ is a solution of $$4 \cos \theta-3 \sin \theta=1$$. Then $$\cos \theta$$ is equal to :

JEE Main 2024 (Online) 5th April Morning Shift
10
If $\tan \mathrm{A}=\frac{1}{\sqrt{x\left(x^2+x+1\right)}}, \tan \mathrm{B}=\frac{\sqrt{x}}{\sqrt{x^2+x+1}}$ and

$\tan \mathrm{C}=\left(x^{-3}+x^{-2}+x^{-1}\right)^{1 / 2}, 0<\mathrm{A}, \mathrm{B}, \mathrm{C}<\frac{\pi}{2}$, then $\mathrm{A}+\mathrm{B}$ is equal to :
JEE Main 2024 (Online) 1st February Morning Shift
11

The number of solutions, of the equation $$e^{\sin x}-2 e^{-\sin x}=2$$, is :

JEE Main 2024 (Online) 31st January Evening Shift
12

For $$\alpha, \beta \in(0, \pi / 2)$$, let $$3 \sin (\alpha+\beta)=2 \sin (\alpha-\beta)$$ and a real number $$k$$ be such that $$\tan \alpha=k \tan \beta$$. Then, the value of $$k$$ is equal to

JEE Main 2024 (Online) 30th January Evening Shift
13

$$96\cos {\pi \over {33}}\cos {{2\pi } \over {33}}\cos {{4\pi } \over {33}}\cos {{8\pi } \over {33}}\cos {{16\pi } \over {33}}$$ is equal to :

JEE Main 2023 (Online) 10th April Morning Shift
14

The value of $$36\left(4 \cos ^{2} 9^{\circ}-1\right)\left(4 \cos ^{2} 27^{\circ}-1\right)\left(4 \cos ^{2} 81^{\circ}-1\right)\left(4 \cos ^{2} 243^{\circ}-1\right)$$ is :

JEE Main 2023 (Online) 8th April Evening Shift
15

If $$\tan 15^\circ + {1 \over {\tan 75^\circ }} + {1 \over {\tan 105^\circ }} + \tan 195^\circ = 2a$$, then the value of $$\left( {a + {1 \over a}} \right)$$ is :

JEE Main 2023 (Online) 30th January Morning Shift
16

The set of all values of $$\lambda$$ for which the equation $${\cos ^2}2x - 2{\sin ^4}x - 2{\cos ^2}x = \lambda $$ has a real solution $$x$$, is :

JEE Main 2023 (Online) 29th January Evening Shift
17

Let $$f(\theta ) = 3\left( {{{\sin }^4}\left( {{{3\pi } \over 2} - \theta } \right) + {{\sin }^4}(3\pi + \theta )} \right) - 2(1 - {\sin ^2}2\theta )$$ and $$S = \left\{ {\theta \in [0,\pi ]:f'(\theta ) = - {{\sqrt 3 } \over 2}} \right\}$$. If $$4\beta = \sum\limits_{\theta \in S} \theta $$, then $$f(\beta )$$ is equal to

JEE Main 2023 (Online) 29th January Morning Shift
18

$$2 \sin \left(\frac{\pi}{22}\right) \sin \left(\frac{3 \pi}{22}\right) \sin \left(\frac{5 \pi}{22}\right) \sin \left(\frac{7 \pi}{22}\right) \sin \left(\frac{9 \pi}{22}\right)$$ is equal to :

JEE Main 2022 (Online) 25th July Evening Shift
19

If cot$$\alpha$$ = 1 and sec$$\beta$$ = $$ - {5 \over 3}$$, where $$\pi < \alpha < {{3\pi } \over 2}$$ and $${\pi \over 2} < \beta < \pi $$, then the value of $$\tan (\alpha + \beta )$$ and the quadrant in which $$\alpha$$ + $$\beta$$ lies, respectively are :

JEE Main 2022 (Online) 28th June Evening Shift
20

$$\alpha = \sin 36^\circ $$ is a root of which of the following equation?

JEE Main 2022 (Online) 27th June Evening Shift
21

The value of $$\cos \left( {{{2\pi } \over 7}} \right) + \cos \left( {{{4\pi } \over 7}} \right) + \cos \left( {{{6\pi } \over 7}} \right)$$ is equal to :

JEE Main 2022 (Online) 27th June Morning Shift
22

$$16\sin (20^\circ )\sin (40^\circ )\sin (80^\circ )$$ is equal to :

JEE Main 2022 (Online) 26th June Evening Shift
23

The value of 2sin (12$$^\circ$$) $$-$$ sin (72$$^\circ$$) is :

JEE Main 2022 (Online) 25th June Evening Shift
24
The value of

$$2\sin \left( {{\pi \over 8}} \right)\sin \left( {{{2\pi } \over 8}} \right)\sin \left( {{{3\pi } \over 8}} \right)\sin \left( {{{5\pi } \over 8}} \right)\sin \left( {{{6\pi } \over 8}} \right)\sin \left( {{{7\pi } \over 8}} \right)$$ is :
JEE Main 2021 (Online) 26th August Evening Shift
25
If $$\tan \left( {{\pi \over 9}} \right),x,\tan \left( {{{7\pi } \over {18}}} \right)$$ are in arithmetic progression and $$\tan \left( {{\pi \over 9}} \right),y,\tan \left( {{{5\pi } \over {18}}} \right)$$ are also in arithmetic progression, then $$|x - 2y|$$ is equal to :
JEE Main 2021 (Online) 27th July Evening Shift
26
If $$\sin \theta + \cos \theta = {1 \over 2}$$, then 16(sin(2$$\theta$$) + cos(4$$\theta$$) + sin(6$$\theta$$)) is equal to :
JEE Main 2021 (Online) 27th July Morning Shift
27
The value of $$\cot {\pi \over {24}}$$ is :
JEE Main 2021 (Online) 25th July Evening Shift
28
If 15sin4$$\alpha$$ + 10cos4$$\alpha$$ = 6, for some $$\alpha$$$$\in$$R, then the value of

27sec6$$\alpha$$ + 8cosec6$$\alpha$$ is equal to :
JEE Main 2021 (Online) 18th March Evening Shift
29
If for x $$\in$$ $$\left( {0,{\pi \over 2}} \right)$$, log10sinx + log10cosx = $$-$$1 and log10(sinx + cosx) = $${1 \over 2}$$(log10 n $$-$$ 1), n > 0, then the value of n is equal to :
JEE Main 2021 (Online) 16th March Morning Shift
30
If 0 < x, y < $$\pi$$ and cosx + cosy $$-$$ cos(x + y) = $${3 \over 2}$$, then sinx + cosy is equal to :
JEE Main 2021 (Online) 25th February Evening Shift
31
If $${e^{\left( {{{\cos }^2}x + {{\cos }^4}x + {{\cos }^6}x + ...\infty } \right){{\log }_e}2}}$$ satisfies the equation t2 - 9t + 8 = 0, then the value of
$${{2\sin x} \over {\sin x + \sqrt 3 \cos x}}\left( {0 < x < {\pi \over 2}} \right)$$ is :
JEE Main 2021 (Online) 24th February Morning Shift
32
If L = sin2$$\left( {{\pi \over {16}}} \right)$$ - sin2$$\left( {{\pi \over {8}}} \right)$$ and
M = cos2$$\left( {{\pi \over {16}}} \right)$$ - sin2$$\left( {{\pi \over {8}}} \right)$$, then :
JEE Main 2020 (Online) 5th September Evening Slot
33
If the equation cos4 $$\theta $$ + sin4 $$\theta $$ + $$\lambda $$ = 0 has real solutions for $$\theta $$, then $$\lambda $$ lies in the interval :
JEE Main 2020 (Online) 2nd September Evening Slot
34
If $$x = \sum\limits_{n = 0}^\infty {{{\left( { - 1} \right)}^n}{{\tan }^{2n}}\theta } $$ and $$y = \sum\limits_{n = 0}^\infty {{{\cos }^{2n}}\theta } $$

for 0 < $$\theta $$ < $${\pi \over 4}$$, then :
JEE Main 2020 (Online) 9th January Evening Slot
35
The value of
$${\cos ^3}\left( {{\pi \over 8}} \right)$$$${\cos}\left( {{3\pi \over 8}} \right)$$+$${\sin ^3}\left( {{\pi \over 8}} \right)$$$${\sin}\left( {{3\pi \over 8}} \right)$$
is :
JEE Main 2020 (Online) 9th January Morning Slot
36
The equation y = sinx sin (x + 2) – sin2 (x + 1) represents a straight line lying in :
JEE Main 2019 (Online) 12th April Morning Slot
37
The value of sin 10º sin30º sin50º sin70º is :-
JEE Main 2019 (Online) 9th April Evening Slot
38
The value of cos210° – cos10°cos50° + cos250° is
JEE Main 2019 (Online) 9th April Morning Slot
39
If cos($$\alpha $$ + $$\beta $$) = 3/5 ,sin ( $$\alpha $$ - $$\beta $$) = 5/13 and 0 < $$\alpha , \beta$$ < $$\pi \over 4$$, then tan(2$$\alpha $$) is equal to :
JEE Main 2019 (Online) 8th April Morning Slot
40
The maximum value of 3cos$$\theta $$ + 5sin $$\left( {\theta - {\pi \over 6}} \right)$$ for any real value of $$\theta $$ is :
JEE Main 2019 (Online) 12th January Morning Slot
41
The value of $$\cos {\pi \over {{2^2}}}.\cos {\pi \over {{2^3}}}\,.....\cos {\pi \over {{2^{10}}}}.\sin {\pi \over {{2^{10}}}}$$ is -
JEE Main 2019 (Online) 10th January Evening Slot
42
For any $$\theta \in \left( {{\pi \over 4},{\pi \over 2}} \right)$$, the expression

$$3{(\cos \theta - \sin \theta )^4}$$$$ + 6{(\sin \theta + \cos \theta )^2} + 4{\sin ^6}\theta $$

equals :
JEE Main 2019 (Online) 9th January Morning Slot
43
If $$5\left( {{{\tan }^2}x - {{\cos }^2}x} \right) = 2\cos 2x + 9$$,

then the value of $$\cos 4x$$ is :
JEE Main 2017 (Offline)
44
If  m and M are the minimum and the maximum values of

4 + $${1 \over 2}$$ sin2 2x $$-$$ 2cos4 x, x $$ \in $$ R, then M $$-$$ m is equal to :
JEE Main 2016 (Online) 9th April Morning Slot
45
Let $$f_k\left( x \right) = {1 \over k}\left( {{{\sin }^k}x + {{\cos }^k}x} \right)$$ where $$x \in R$$ and $$k \ge \,1.$$
Then $${f_4}\left( x \right) - {f_6}\left( x \right)\,\,$$ equals :
JEE Main 2014 (Offline)
46
The expression $${{\tan {\rm A}} \over {1 - \cot {\rm A}}} + {{\cot {\rm A}} \over {1 - \tan {\rm A}}}$$ can be written as:
JEE Main 2013 (Offline)
47
If $$A = {\sin ^2}x + {\cos ^4}x,$$ then for all real $$x$$:
AIEEE 2011
48
Let $$\cos \left( {\alpha + \beta } \right) = {4 \over 5}$$ and $$\sin \,\,\,\left( {\alpha - \beta } \right) = {5 \over {13}},$$ where $$0 \le \alpha ,\,\beta \le {\pi \over 4}.$$
Then $$tan\,2\alpha $$ =
AIEEE 2010
49
Let A and B denote the statements

A: $$\cos \alpha + \cos \beta + \cos \gamma = 0$$

B: $$\sin \alpha + \sin \beta + \sin \gamma = 0$$

If $$\cos \left( {\beta - \gamma } \right) + \cos \left( {\gamma - \alpha } \right) + \cos \left( {\alpha - \beta } \right) = - {3 \over 2},$$ then:

AIEEE 2009
50
If $$0 < x < \pi $$ and $$\cos x + \sin x = {1 \over 2},$$ then $$\tan x$$ is :
AIEEE 2006
51
If $$u = \sqrt {{a^2}{{\cos }^2}\theta + {b^2}{{\sin }^2}\theta } + \sqrt {{a^2}{{\sin }^2}\theta + {b^2}{{\cos }^2}\theta } $$

then the difference between the maximum and minimum values of $${u^2}$$ is given by :
AIEEE 2004
52
Let $$\alpha ,\,\beta $$ be such that $$\pi < \alpha - \beta < 3\pi $$.
If $$sin{\mkern 1mu} \alpha + \sin \beta = - {{21} \over {65}}$$ and $$\cos \alpha + \cos \beta = - {{27} \over {65}}$$ then the value of $$\cos {{\alpha - \beta } \over 2}$$ :
AIEEE 2004

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