If $\;0^\circ < {\rm{\theta }} < 180^\circ ,\;\;{\rm{then}}\,\,\sqrt {2\,\, + \,\,\sqrt {2\,\, + \,\,\sqrt {2\,\, + \,\,... + \sqrt {2\,\,\left( {1 + \cos \,\,{\rm{\theta }}} \right),} } } } $ then being n number of ${\rm{2'}}\,{\rm{s}}$, is equal to

$2\,\,\cos \,\,\left( {\frac{{\rm{\theta }}}{{{2^n}}}} \right)$

$2\,\,\cos \,\,\left( {\frac{{\rm{\theta }}}{{{2^{n - 1}}}}} \right)$

$2\,\,\cos \,\,\left( {\frac{{\rm{\theta }}}{{{2^{n + 1}}}}} \right)$

If $\tan x=\frac{b}{a}$, then $\sqrt{\frac{a+b}{a-b}}+\sqrt{\frac{a-b}{a+b}}=$

$\frac{2 \sin x}{\sqrt{\sin 2 x}}$

$\frac{2 \cos x}{\sqrt{\sin 2 x}}$

$\frac{2 \sin x}{\sqrt{\cos 2 x}}$

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JEE • MATHEMATICS • MULTIPLE AND SUBMULTIPLE ANGLES

Multiple and submultiple angles practice · Mixed difficulty

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Showing 61-90 of 122 questions

61 Easy

61 of 122

Application +4 | -1

\;x\,\,\cos \,\,\mathrm{\theta } = y\,\,\cos \left( {\mathrm{\theta } + \frac{{2\pi }}{3}} \right) = \mathrm{z}\,\,\cos \left( {\mathrm{\theta } + \frac{{4\pi }}{3}} \right)\;

, then the value of

\frac{1}{x} + \frac{1}{y} + \frac{1}{\mathrm{z}}

is equal to

View answer and explanation

Correct option: C

Option C (0) is correct because the sum of the cosines with the given phase shifts equals zero, leading to the result 1/x + 1/y + 1/z = 0.

62 Easy

62 of 122

Application +4 | -1

\mathrm{cosec}\;\,\,\theta = \frac{{p + q}}{{p - q}}

, then

\cot \,\,\left( {\pi /4 + \theta /2} \right) =

View answer and explanation

Correct option: B

\sqrt {\frac{q}{p}}

Option B,

\sqrt {\frac{q}{p}}

, is correct because it is the simplified form of

\cot \left( {\pi /4 + \theta /2} \right)

after substituting the given value of

\mathrm{cosec}\;\,\theta

and using trigonometric identities.

63 Medium

63 of 122

Analyzing +4 | -1

x = y\,\,\cos \,\,\frac{{2\,\pi }}{3} = z\,\,\cos \frac{{4\,\pi }}{3}

, then

xy + yz + zx =

View answer and explanation

Correct option: B

Option B (0) is correct because when x, y, and z are expressed in terms of a common variable, the expression xy + yz + zx simplifies to 0.

64 Medium

64 of 122

Analyzing +4 | -1

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)

View answer and explanation

Correct option: D

Option D, 36, is the correct value of the given expression. This is found by using trigonometric identities to simplify the expression.

65 Easy

65 of 122

Application +4 | -1

The value of the series

\cos \,\,12^\circ + \cos \,\,84^\circ + \cos \,\,132^\circ + \cos \,\,156^\circ

View answer and explanation

Correct option: D

- \frac{1}{2}

Option D is correct because the sum of the cosine series simplifies to -1/2 using trigonometric identities.

66 Medium

66 of 122

Application +4 | -1

\frac{{{a^2} + 1}}{{2a}} = \cos \,\,\theta

, then

\frac{{{a^6} + 1}}{{2{a^3}}} =

View answer and explanation

Correct option: D

\cos \,\,3\theta

Option D (

\cos \,3\theta

) is correct because by manipulating the given equation and using trigonometric identities, we can find that

\frac{{a^6 + 1}}{{2{a^3}}} = \cos \,3\theta

67 Easy

67 of 122

Application +4 | -1

The value of

\mathrm{\;\;}\left( {1 + \mathrm{cos}\,\,\frac{\mathrm{\pi }}{8}} \right)\,\,\left( {1 + \mathrm{cos}\,\,\frac{{3\mathrm{\pi }}}{8}} \right)\,\,\left( {1 + \mathrm{cos}\,\,\frac{{5\mathrm{\pi }}}{8}} \right)\,\,\left( {1 + \mathrm{cos}\,\,\frac{{7\mathrm{\pi }}}{8}} \right)\mathrm{\;}

is equal to

View answer and explanation

Correct option: C

\frac{1}{8}

Option D (

n\pi \\ + \\ \frac{\pi }{4}

) is correct because by simplifying the equation, we get

\tan \theta = 1

, which gives the general solution

\theta = n\pi + \frac{\pi}{4}

68 Hard

68 of 122

Analyzing +4 | -1

\cos \,\,\frac{x}{2}\,\,.\,\,\cos \,\,\frac{x}{{{2^2}}} \ldots ..\cos \,\,\frac{x}{{{2^n}}} = \frac{{\sin \,\,x}}{{{2^n}\,\,\sin \;\,\,\frac{x}{{{2^n}}}}}

, then

\frac{1}{2}\,\,\tan \,\,\frac{x}{2} + \frac{1}{{{2^2}}}\,\,\tan \,\,\frac{x}{{{2^2}}} + ... + \frac{1}{{{2^n}}}\,\,\tan \,\,\frac{x}{{{2^n}}}

View answer and explanation

Correct option: B

\frac{1}{{{2^n}}}\,\,\cot \,\,\left( {\frac{x}{{{2^n}}}} \right) - \cot \,\,x

Option B (1/2^n cot(x/2^n) - cot x) is correct because by repeatedly using the identity tan x/2 = cot x/2 - 2cot x, the expression simplifies to the given form.

69 Easy

69 of 122

Application +4 | -1

The value of

{\cos ^2}\,\,\frac{\pi }{{16}} + {\cos ^2}\,\,\frac{{3\pi }}{{16}} + {\cos ^2}\,\,\frac{{5\pi }}{{16}} + {\cos ^2}\,\,\frac{{7\pi }}{{16}}

View answer and explanation

Correct option: A

Option A (2) is correct because using trigonometric identities, the expression simplifies to 2.

70 Easy

70 of 122

Analyzing +4 | -1

\left( {1 + \cos \,\,\frac{\pi }{8}} \right)\,\,\left( {1 + \cos \,\,\frac{{3\,\pi }}{8}} \right)\,\,\left( {1 + \cos \,\,\frac{{5\,\pi }}{8}} \right)\,\,\left( {1 + \cos \,\,\frac{{7\pi }}{8}} \right)

is equal to

View answer and explanation

Correct option: C

\frac{1}{8}

Option C (

\frac{1}{8}

) is correct because the given expression simplifies to

\frac{1}{8}

using trigonometric identities and properties of cosine.

71 Medium

71 of 122

Application +4 | -1

\tan \mathrm{\theta } + \tan \left( {\frac{\pi }{3} + \mathrm{\theta }} \right) + \tan \left( { - \frac{\pi }{3} + \mathrm{\theta }} \right) = a\tan 3\mathrm{\theta },

then

a

is equal to

View answer and explanation

Correct option: C

Option C (3) is correct because by simplifying the given trigonometric expression, we find that

a = 3

72 Medium

72 of 122

Application +4 | -1

\;0^\circ < \mathrm{\theta } < 180^\circ ,\;\;\mathrm{then}\,\,\sqrt {2\,\, + \,\,\sqrt {2\,\, + \,\,\sqrt {2\,\, + \,\,... + \sqrt {2\,\,\left( {1 + \cos \,\,\mathrm{\theta }} \right),} } } }

then being n number of

\mathrm{2'}\,\mathrm{s}

, is equal to

View answer and explanation

Correct option: A

2\,\,\cos \,\,\left( {\frac{\mathrm{\theta }}{{{2^n}}}} \right)

Option A (

2\cos \left( {\frac{\mathrm{\theta }}{{{{2^n}}}}} \right)

) is correct because the nested square roots simplify using trigonometric identities to this expression.

73 Easy

73 of 122

Application +4 | -1

\tan x=\frac{b}{a}

, then

\sqrt{\frac{a+b}{a-b}}+\sqrt{\frac{a-b}{a+b}}=

View answer and explanation

Correct option: B

\frac{2 \cos x}{\sqrt{\cos 2 x}}

Option B (

\frac{{2 \cos x}}{{\sqrt{{\cos 2 x}}}}

) is correct because, after using componendo and dividendo and simplifying, the expression equals

\frac{{2 \cos x}}{{\sqrt{{\cos 2 x}}}}

74 Easy

74 of 122

Application +4 | -1

The value of

\mathrm{\;tan}\,\,20^\circ \,\,\tan \,\,40^\circ \,\,\tan \,\,60^\circ \,\,\tan \,\,80^\circ \mathrm{\;}

is equal to

View answer and explanation

Correct option: C

Option C (3) is correct because the product of the tangent values simplifies to 3 using trigonometric identities.

75 Easy

75 of 122

Analyzing +4 | -1

\tan \,\,x = \frac{{2b}}{{a - c}},\,\;a \ne c

and

y = a\,\,{\cos ^2}\,x\,\, + \,\,2b\,\,\sin \,\,x\,\,\cos \,\,x\,\, + \,\,c\,\,{\sin ^2}\,x

z = a\,\,{\sin ^2}\,x\,\, - \,\,2b\,\,\sin \,\,x\,\,\cos \,\,x\,\, + \,\,c\,\,{\cos ^2}\,x

, then

View answer and explanation

Correct option: C

y - z = a - c

Option C (

y - z = a - c

) is correct because simplifying the expression

y - z

using trigonometric identities and the given condition

\tan x = \frac{{2b}}{{a - c}}

leads to the result

a - c

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SEQUENCE OF SERIES 1764 MCQs DEFINITE INTEGRATION 1443 MCQs MATRICES 1363 MCQs INDEFINITE INTEGRATION 1179 MCQs PROBABILITY 1146 MCQs COMPLEX NUMBERS 980 MCQs

76 Easy

76 of 122

Understanding +4 | -1

1+\cos ^2 2 A

is equal to

View answer and explanation

Correct option: C

2\left(\cos ^4 \mathrm{~A}+\sin ^4 \mathrm{~A}\right)

Option C (

2(\cos ^4 A+\sin ^4 A)

) is correct because it is the simplified form of the given expression using trigonometric identities.

77 Easy

77 of 122

Application +4 | -1

\cos A=3 / 4

then value of

32 \sin (A / 2) \sin (5 \mathrm{~A} / 2)

is equal to

View answer and explanation

Correct option: C

Option C (11) is correct because it is the result of applying the product-to-sum trigonometric identity and simplifying using the given value of

\cos A

78 Medium

78 of 122

Application +4 | -1

The value of

\cos ^4\left(\frac{\pi}{8}\right)+\cos ^4\left(\frac{3 \pi}{8}\right)+

\cos ^4\left(\frac{5 \pi}{8}\right)+\cos ^4\left(\frac{7 \pi}{8}\right)

is equal to

View answer and explanation

Correct option: C

3 / 2

Option C (

3/2

) is the correct answer. The expression simplifies using trigonometric identities and the property that

\cos(\pi - x) = -\cos(x)

and

\cos(\frac{\pi}{2} - x) = \sin(x)

79 Easy

79 of 122

Application +4 | -1

\frac{{\mathrm{sin}\,\,\mathrm{\;}\left( {x + y} \right)}}{{\sin \,\,\;\left( {x - y} \right)}} = \frac{{a + b}}{{a - b}}

, then

\frac{{\tan \,\,x}}{{\tan \,\,y}}

is equal to

View answer and explanation

Correct option: B

\frac{a}{b}

Option B (

\frac{a}{b}

) is correct because by applying the componendo and dividendo rule and trigonometric identities, the ratio of tangents simplifies to

\frac{a}{b}

80 Medium

80 of 122

Analyzing +4 | -1

The number of all possible ordered pairs

\left( {x,\,y} \right),\;\,x,\,\,y \in R

satisfying the system of equations

x + y = \frac{{2\;\pi }}{3},\,\,\cos \,\,x + \cos \,\,y = \frac{3}{2},\;\,\mathrm{is}

View answer and explanation

Correct option: A

Option A (0) is correct because the given system of equations has no solution, as the second equation leads to an impossible value for cosine.

81 Medium

81 of 122

Application +4 | -1

\tan x = \frac{b}{a}

, then the value of

a\,\,\cos \,\,2x + b\,\,\sin \,\,2x

View answer and explanation

Correct option: A

a

Option A (

a

) is correct because substituting the given value of

\tan x

and simplifying the expression results in

a

82 Easy

82 of 122

Application +4 | -1

5\cos \,\,2\theta + 2{\cos ^2}\frac{\theta }{2} + 1 = 0,\; - \pi < \theta < \pi

, then

\theta

is equal to

View answer and explanation

Correct option: D

\frac{\pi }{3},\;\,\pi - {\cos ^{ - 1}}\left( {\frac{3}{5}} \right)

Option D (

\frac{\pi }{3},\;\,\pi - {\cos ^{ - 1}}\left( {\frac{3}{5}} \right)

) is correct because it represents the solutions obtained by solving the given trigonometric equation within the specified interval.

83 Easy

83 of 122

Analyzing +4 | -1

2\,\,\cos \,\,\frac{A}{2} = \sqrt {1 + \sin \,\,A} + \sqrt {1 - \sin \,\,A}

, then

\frac{A}{2}

lies between,

\left( {n \in Z} \right)

View answer and explanation

Correct option: B

2n\;\pi - \frac{\pi }{4}

and

2n\;\pi + \frac{\pi }{4}

Option C (

\cot B

) is correct because simplifying the given expression using trigonometric identities leads to this result.

84 Medium

84 of 122

Application +4 | -1

The value of

{\cos ^2}10^\circ - \cos 10^\circ \cos 50^\circ + {\cos ^2}50^\circ

View answer and explanation

Correct option: D

\frac{3}{4}

Option D (

\frac{{3}}{{4}}

) is correct because the expression simplifies to this value using trigonometric identities.

85 Easy

85 of 122

Application +4 | -1

The value of

\sqrt{2+\sqrt{2+\sqrt{2+2 \cos 8 \theta}}}

, where

0 < \theta < \frac{\pi}{8}

, is

View answer and explanation

Correct option: A

2 \cos \theta

Option A (

2 \cos \theta

) is correct because the nested square roots simplify to

2 \cos \theta

using the double angle formula for cosine.

86 Easy

86 of 122

Application +4 | -1

The value of

\cos \,\,\frac{\pi }{{15}}\,\,\cos \,\,\frac{{2\,\pi }}{{15}}\,\,\cos \,\,\frac{{3\,\pi }}{{15}}\,\,\cos \,\,\frac{{4\,\pi }}{{15}}\,\,\cos \,\,\frac{{5\,\pi }}{{15}}\,\,\cos \,\,\frac{{6\,\pi }}{{15}}\,\,\cos \,\,\frac{{7\pi }}{{15}}

, is

View answer and explanation

Correct option: A

1/128

Option A (

1/128

) is correct because the expression can be simplified using the identity

\prod_{k=1}^{n-1} \cos(\frac{k\pi}{2n}) = \frac{\sqrt{n}}{2^{n-1}}

87 Easy

87 of 122

Application +4 | -1

\cos \,\,\left( {\theta - \alpha } \right),\,\,\cos \,\,\theta ,\,\,\cos \,\,\left( {\theta + \alpha } \right)

are in H.P., then

\cos \,\,\theta \,\,\sec \,\,\left( {\frac{\alpha }{2}} \right)

is equal to

View answer and explanation

Correct option: B

\pm \;\,\sqrt 2

Option B (

\pm \;\,\sqrt 2

) is correct because by using the harmonic progression property and trigonometric identities, the expression

\cos \,\theta \,\,\sec \,\left( {\frac{\alpha }{2}} \right)

simplifies to

\pm \;\,\sqrt 2

88 Easy

88 of 122

Analyzing +4 | -1

\sin x + \sin y = 3\left( {\cos y - \cos x} \right)

, then the value of

\frac{{\sin 3x}}{{\sin 3y}}

View answer and explanation

Correct option: B

- 1

Option B (-1) is correct because by simplifying the given equation, we find a relationship between x and y, allowing us to evaluate the expression.

89 Easy

89 of 122

Analyzing +4 | -1

The value of

\sin \,\,\frac{{15\,\pi }}{{32}}\,\,\sin \,\,\frac{{7\pi }}{{16}}\,\,\sin \,\,\frac{{3\,\pi }}{8}

View answer and explanation

Correct option: A

\frac{1}{{8\,\sqrt 2 \,\,\cos \,\,\left( {\frac{{15\pi }}{{32}}} \right)}}

Option A (

\frac{1}{{8\sqrt 2 \cos \left( {\frac{{15\pi }}{{32}}} \right)}}

) is correct because it is derived by repeatedly applying the double angle formula for sine.

90 Medium

90 of 122

Application +4 | -1

\cot \theta=\frac{\sqrt{m}+\sqrt{n}}{\sqrt{m}-\sqrt{n}}

, then

(m+n) \cos 2 \theta

is equal to

View answer and explanation

Correct option: A

2 \sqrt{m n}

Option A (

2 \sqrt{mn}

) is correct because by using the given equation and applying the double angle formula for cosine, the expression simplifies to

2 \sqrt{mn}

Revision notes and key takeaways

Review MULTIPLE AND SUBMULTIPLE ANGLES in your textbook or class notes, then use this set to check your understanding.

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