JEE Main
Mathematics
Trigonometric Ratio and Identites
Previous Years Questions

$$96\cos {\pi \over {33}}\cos {{2\pi } \over {33}}\cos {{4\pi } \over {33}}\cos {{8\pi } \over {33}}\cos {{16\pi } \over {33}}$$ is equal to
The value of $$36\left(4 \cos ^{2} 9^{\circ}-1\right)\left(4 \cos ^{2} 27^{\circ}-1\right)\left(4 \cos ^{2} 81^{\circ}-1\right)\left(4 \cos ^{2} 243^{... If$$\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}} ...
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 :...
Let $$f(\theta ) = 3\left( {{{\sin }^4}\left( {{{3\pi } \over 2} - \theta } \right) + {{\sin }^4}(3\pi + \theta )} \right) - 2(1 - {\sin ^2}2\theta )...$$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...
If cot$$\alpha$$ = 1 and sec$$\beta$$ = $$- {5 \over 3}$$, where $$\pi ...$$\alpha = \sin 36^\circ $$is a root of which of the following equation? The value of$$\cos \left( {{{2\pi } \over 7}} \right) + \cos \left( {{{4\pi } \over 7}} \right) + \cos \left( {{{6\pi } \over 7}} \right)$$is equal ...$$16\sin (20^\circ )\sin (40^\circ )\sin (80^\circ )$$is equal to : The value of 2sin (12$$^\circ$$)$$-$$sin (72$$^\circ$$) is : The value of$$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 ...
If $$\tan \left( {{\pi \over 9}} \right),x,\tan \left( {{{7\pi } \over {18}}} \right)$$ are in arithmetic progression and $$\tan \left( {{\pi \over ... If$$\sin \theta + \cos \theta = {1 \over 2}$$, then 16(sin(2$$\theta$$) + cos(4$$\theta$$) + sin(6$$\theta$$)) is equal to : The value of$$\cot {\pi \over {24}}$$is : If 15sin4$$\alpha$$+ 10cos4$$\alpha$$= 6, for some$$\alpha\in$$R, then the value of 27sec6$$\alpha$$+ 8cosec6$$\alpha$$is equal to :... If for x$$\in\left( {0,{\pi \over 2}} \right)$$, log10sinx + log10cosx =$$-$$1 and log10(sinx + cosx) =$${1 \over 2}$$(log10 n$$-$$1), n &g... If 0 < x, y <$$\pi$$and cosx + cosy$$-$$cos(x + y) =$${3 \over 2}$$, then sinx + cosy is equal to : If$${e^{\left( {{{\cos }^2}x + {{\cos }^4}x + {{\cos }^6}x + ...\infty } \right){{\log }_e}2}}$$satisfies the equation t2 - 9t + 8 = 0, then the val... If L = sin2$$\left( {{\pi \over {16}}} \right)$$- sin2$$\left( {{\pi \over {8}}} \right)$$and M = cos2$$\left( {{\pi \over {16}}} \right)$$- sin... If the equation cos4$$\theta $$+ sin4$$\theta $$+$$\lambda $$= 0 has real solutions for$$\theta $$, then$$\lambda $$lies in the interval :... If$$x = \sum\limits_{n = 0}^\infty {{{\left( { - 1} \right)}^n}{{\tan }^{2n}}\theta } $$and$$y = \sum\limits_{n = 0}^\infty {{{\cos }^{2n}}\thet...
The value of $${\cos ^3}\left( {{\pi \over 8}} \right)$$$${\cos}\left( {{3\pi \over 8}} \right)$$+$${\sin ^3}\left( {{\pi \over 8}} \right)$$$${\si... The equation y = sinx sin (x + 2) – sin2 (x + 1) represents a straight line lying in : The value of sin 10º sin30º sin50º sin70º is :- The value of cos210° – cos10°cos50° + cos250° is If cos($$\alpha $$+$$\beta $$) = 3/5 ,sin ($$\alpha $$-$$\beta $$) = 5/13 and 0 <$$\alpha , \beta$$<$$\pi \over 4$$, then tan(2$$\alpha ...
The maximum value of 3cos$$\theta$$ + 5sin $$\left( {\theta - {\pi \over 6}} \right)$$ for any real value of $$\theta$$ is :
The value of $$\cos {\pi \over {{2^2}}}.\cos {\pi \over {{2^3}}}\,.....\cos {\pi \over {{2^{10}}}}.\sin {\pi \over {{2^{10}}}}$$ is -
For any $$\theta \in \left( {{\pi \over 4},{\pi \over 2}} \right)$$, the expression $$3{(\cos \theta - \sin \theta )^4}$$$$+ 6{(\sin \theta + ... If$$5\left( {{{\tan }^2}x - {{\cos }^2}x} \right) = 2\cos 2x + 9$$, then the value of$$\cos 4x$$is : If m and M are the minimum and the maximum values of 4 +$${1 \over 2}$$sin2 2x$$-$$2cos4 x, x$$ \in $$R, then M$$-$$m is equal to :... Let$$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...
The expression $${{\tan {\rm A}} \over {1 - \cot {\rm A}}} + {{\cot {\rm A}} \over {1 - \tan {\rm A}}}$$ can be written as:
If $$A = {\sin ^2}x + {\cos ^4}x,$$ then for all real $$x$$:
Let $$\cos \left( {\alpha + \beta } \right) = {4 \over 5}$$ and $$\sin \,\,\,\left( {\alpha - \beta } \right) = {5 \over {13}},$$ where $$0 \le \alp... Let A and B denote the statements A:$$\cos \alpha + \cos \beta + \cos \gamma = 0$$B:$$\sin \alpha + \sin \beta + \sin \gamma = 0$$If$$\cos...
If $$0 < x < \pi$$ and $$\cos x + \sin x = {1 \over 2},$$ then $$\tan x$$ is :
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...
Let $$\alpha ,\,\beta$$ be such that $$\pi < \alpha - \beta < 3\pi$$. If $$sin{\mkern 1mu} \alpha + \sin \beta = - {{21} \over {65}}$$ a...

## Numerical

The value of $$\tan 9^{\circ}-\tan 27^{\circ}-\tan 63^{\circ}+\tan 81^{\circ}$$ is __________.
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 $$16 + {\... The number of integral values of 'k' for which the equation$$3\sin x + 4\cos x = k + 1$$has a solution, k$$\in$$R is ___________. If$${{\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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