# Trigonometric Ratio and Identites · Mathematics · JEE Main

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JEE Main 2024 (Online) 8th April Evening Shift
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 $$a, b, c$$ a...
JEE Main 2024 (Online) 8th April Morning Shift
If $$\sin x=-\frac{3}{5}$$, where $$\pi JEE Main 2024 (Online) 5th April Morning Shift Suppose$$\theta \in\left[0, \frac{\pi}{4}\right]$$is a solution of$$4 \cos \theta-3 \sin \theta=1$$. Then$$\cos \theta$$is equal to : JEE Main 2024 (Online) 1st February Morning Shift If \tan \mathrm{A}=\frac{1}{\sqrt{x\left(x^2+x+1\right)}}, \tan \mathrm{B}=\frac{\sqrt{x}}{\sqrt{x^2+x+1}} and \tan \mathrm{C}=\left(x^{-3}+x^{-2}+... JEE Main 2024 (Online) 31st January Evening Shift The number of solutions, of the equation$$e^{\sin x}-2 e^{-\sin x}=2$$, is : JEE Main 2024 (Online) 30th January Evening Shift For$$\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 \...
JEE Main 2023 (Online) 10th April Morning Shift
$$96\cos {\pi \over {33}}\cos {{2\pi } \over {33}}\cos {{4\pi } \over {33}}\cos {{8\pi } \over {33}}\cos {{16\pi } \over {33}}$$ is equal to :
JEE Main 2023 (Online) 8th April Evening Shift
The value of $$36\left(4 \cos ^{2} 9^{\circ}-1\right)\left(4 \cos ^{2} 27^{\circ}-1\right)\left(4 \cos ^{2} 81^{\circ}-1\right)\left(4 \cos ^{2} 243^{... JEE Main 2023 (Online) 30th January Morning Shift If$$\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}} ...
JEE Main 2023 (Online) 29th January Evening Shift
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 Morning Shift
Let $$f(\theta ) = 3\left( {{{\sin }^4}\left( {{{3\pi } \over 2} - \theta } \right) + {{\sin }^4}(3\pi + \theta )} \right) - 2(1 - {\sin ^2}2\theta )... JEE Main 2022 (Online) 25th July Evening Shift$$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 \le...
JEE Main 2022 (Online) 28th June Evening Shift
If cot$$\alpha$$ = 1 and sec$$\beta$$ = $$- {5 \over 3}$$, where $$\pi ... JEE Main 2022 (Online) 27th June Evening Shift$$\alpha = \sin 36^\circ $$is a root of which of the following equation? JEE Main 2022 (Online) 27th June Morning Shift The value of$$\cos \left( {{{2\pi } \over 7}} \right) + \cos \left( {{{4\pi } \over 7}} \right) + \cos \left( {{{6\pi } \over 7}} \right)$$is equal ... JEE Main 2022 (Online) 26th June Evening Shift$$16\sin (20^\circ )\sin (40^\circ )\sin (80^\circ )$$is equal to : JEE Main 2022 (Online) 25th June Evening Shift The value of 2sin (12$$^\circ$$)$$-$$sin (72$$^\circ$$) is : JEE Main 2021 (Online) 26th August Evening Shift The value of$$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 ...
JEE Main 2021 (Online) 27th July Evening Shift
If $$\tan \left( {{\pi \over 9}} \right),x,\tan \left( {{{7\pi } \over {18}}} \right)$$ are in arithmetic progression and $$\tan \left( {{\pi \over ... JEE Main 2021 (Online) 27th July Morning Shift If$$\sin \theta + \cos \theta = {1 \over 2}$$, then 16(sin(2$$\theta$$) + cos(4$$\theta$$) + sin(6$$\theta$$)) is equal to : JEE Main 2021 (Online) 25th July Evening Shift The value of$$\cot {\pi \over {24}}$$is : JEE Main 2021 (Online) 18th March Evening Shift If 15sin4$$\alpha$$+ 10cos4$$\alpha$$= 6, for some$$\alpha\in$$R, then the value of 27sec6$$\alpha$$+ 8cosec6$$\alpha$$is equal to :... JEE Main 2021 (Online) 16th March Morning Shift If for x$$\in\left( {0,{\pi \over 2}} \right)$$, log10sinx + log10cosx =$$-$$1 and log10(sinx + cosx) =$${1 \over 2}$$(log10 n$$-$$1), n &g... JEE Main 2021 (Online) 25th February Evening Shift If 0 < x, y <$$\pi$$and cosx + cosy$$-$$cos(x + y) =$${3 \over 2}$$, then sinx + cosy is equal to : JEE Main 2021 (Online) 24th February Morning Shift If$${e^{\left( {{{\cos }^2}x + {{\cos }^4}x + {{\cos }^6}x + ...\infty } \right){{\log }_e}2}}$$satisfies the equation t2 - 9t + 8 = 0, then the val... JEE Main 2020 (Online) 5th September Evening Slot If L = sin2$$\left( {{\pi \over {16}}} \right)$$- sin2$$\left( {{\pi \over {8}}} \right)$$and M = cos2$$\left( {{\pi \over {16}}} \right)$$- sin... JEE Main 2020 (Online) 2nd September Evening Slot If the equation cos4$$\theta $$+ sin4$$\theta $$+$$\lambda $$= 0 has real solutions for$$\theta $$, then$$\lambda $$lies in the interval :... JEE Main 2020 (Online) 9th January Evening Slot If$$x = \sum\limits_{n = 0}^\infty {{{\left( { - 1} \right)}^n}{{\tan }^{2n}}\theta } $$and$$y = \sum\limits_{n = 0}^\infty {{{\cos }^{2n}}\thet...
JEE Main 2020 (Online) 9th January Morning Slot
The value of $${\cos ^3}\left( {{\pi \over 8}} \right)$$$${\cos}\left( {{3\pi \over 8}} \right)$$+$${\sin ^3}\left( {{\pi \over 8}} \right)$$$${\si... JEE Main 2019 (Online) 12th April Morning Slot The equation y = sinx sin (x + 2) – sin2 (x + 1) represents a straight line lying in : JEE Main 2019 (Online) 9th April Evening Slot The value of sin 10º sin30º sin50º sin70º is :- JEE Main 2019 (Online) 9th April Morning Slot The value of cos210° – cos10°cos50° + cos250° is JEE Main 2019 (Online) 8th April Morning Slot If cos($$\alpha $$+$$\beta $$) = 3/5 ,sin ($$\alpha $$-$$\beta $$) = 5/13 and 0 <$$\alpha , \beta$$<$$\pi \over 4$$, then tan(2$$\alpha ...
JEE Main 2019 (Online) 12th January Morning Slot
The maximum value of 3cos$$\theta$$ + 5sin $$\left( {\theta - {\pi \over 6}} \right)$$ for any real value of $$\theta$$ is :
JEE Main 2019 (Online) 10th January Evening Slot
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) 9th January Morning Slot
For any $$\theta \in \left( {{\pi \over 4},{\pi \over 2}} \right)$$, the expression $$3{(\cos \theta - \sin \theta )^4}$$$$+ 6{(\sin \theta + ... JEE Main 2017 (Offline) If$$5\left( {{{\tan }^2}x - {{\cos }^2}x} \right) = 2\cos 2x + 9$$, then the value of$$\cos 4x$$is : JEE Main 2016 (Online) 9th April Morning Slot If m and M are the minimum and the maximum values of 4 +$${1 \over 2}$$sin2 2x$$-$$2cos4 x, x$$ \in $$R, then M$$-$$m is equal to :... JEE Main 2014 (Offline) Let$$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...
JEE Main 2013 (Offline)
The expression $${{\tan {\rm A}} \over {1 - \cot {\rm A}}} + {{\cot {\rm A}} \over {1 - \tan {\rm A}}}$$ can be written as:
AIEEE 2011
If $$A = {\sin ^2}x + {\cos ^4}x,$$ then for all real $$x$$:
AIEEE 2010
Let $$\cos \left( {\alpha + \beta } \right) = {4 \over 5}$$ and $$\sin \,\,\,\left( {\alpha - \beta } \right) = {5 \over {13}},$$ where $$0 \le \alp... AIEEE 2009 Let A and B denote the statements A:$$\cos \alpha + \cos \beta + \cos \gamma = 0$$B:$$\sin \alpha + \sin \beta + \sin \gamma = 0$$If$$\cos...
AIEEE 2006
If $$0 < x < \pi$$ and $$\cos x + \sin x = {1 \over 2},$$ then $$\tan x$$ is :
AIEEE 2004
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 differen...
AIEEE 2004
Let $$\alpha ,\,\beta$$ be such that $$\pi < \alpha - \beta < 3\pi$$. If $$sin{\mkern 1mu} \alpha + \sin \beta = - {{21} \over {65}}$$ a...

## Numerical

JEE Main 2024 (Online) 27th January Morning Shift
Let the set of all $a \in \mathbf{R}$ such that the equation $\cos 2 x+a \sin x=2 a-7$ has a solution be $[p, q]$ and \$r=\tan 9^{\circ}-\tan 27^{\circ...
JEE Main 2023 (Online) 6th April Evening Shift
The value of $$\tan 9^{\circ}-\tan 27^{\circ}-\tan 63^{\circ}+\tan 81^{\circ}$$ is __________.
JEE Main 2022 (Online) 26th June Morning Shift
If $${\sin ^2}(10^\circ )\sin (20^\circ )\sin (40^\circ )\sin (50^\circ )\sin (70^\circ ) = \alpha - {1 \over {16}}\sin (10^\circ )$$, then $$16 + {\... JEE Main 2021 (Online) 26th February Morning Shift The number of integral values of 'k' for which the equation$$3\sin x + 4\cos x = k + 1$$has a solution, k$$\in$$R is ___________. JEE Main 2020 (Online) 8th January Evening Slot If$${{\sqrt 2 \sin \alpha } \over {\sqrt {1 + \cos 2\alpha } }} = {1 \over 7}$$and$$\sqrt {{{1 - \cos 2\beta } \over 2}} = {1 \over {\sqrt {10} }}...
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