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 $\cos A + \cos B + \cos C = 0$, then $\cos \,\,3A + \cos \,\,3B + \cos \,\,3C$ is equal to

$8\,\,{\cos ^3}A\,\,{\cos ^3}B\,\,{\cos ^3}C$

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JEE • MATHEMATICS • MULTIPLE AND SUBMULTIPLE ANGLES

Multiple and submultiple angles practice · Medium

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Showing 31-52 of 52 questions

31 Medium

31 of 52

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

32 Medium

32 of 52

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.

33 Medium

33 of 52

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)

34 Medium

34 of 52

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.

35 Medium

35 of 52

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

36 Medium

36 of 52

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.

37 Medium

37 of 52

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}

38 Medium

38 of 52

Application +4 | -1

The value of

16 \cos \left(\frac{2 \pi}{15}\right) \cdot \cos \left(\frac{4 \pi}{15}\right) \cos \left(\frac{8 \pi}{15}\right) \cdot \cos \left(\frac{16 \pi}{15}\right)

View answer and explanation

Correct option: A

Option A (1) is correct because the expression can be simplified using the identity

2 \sin x \cos x = \sin 2x

repeatedly.

39 Medium

39 of 52

Application +4 | -1

The minimum and maximum values of

a b \sin x+b \sqrt{\left(1-a^2\right)} \cos x+c(|a|<1, b>0)

respectively are

View answer and explanation

Correct option: C

\{c-b, b+c\}

Option C ({

c-b, b+c

}) is correct because by rewriting the expression and using trigonometric identities, the minimum and maximum values can be found.

40 Medium

40 of 52

Application +4 | -1

2 \cos \theta+\sin \theta=1

, then

7 \cos \theta+6 \sin \theta

equals

View answer and explanation

Correct option: D

2 or 6

Option D (2 or 6) is correct because by using trigonometric substitutions and solving the resulting quadratic equation, we find two possible values for the expression

7 \cos \theta+6 \sin \theta

41 Medium

41 of 52

Application +4 | -1

\sin \beta

is the geometric mean between

\sin \alpha

and

\cos \alpha

, then

\cos 2 \beta

is equal to

View answer and explanation

Correct option: A

2 \sin ^2\left(\frac{\pi}{4}-\alpha\right)

Option A (

2 \sin ^2\left(\frac{{\pi}}{{4}}-\alpha\right)

) is correct because by using the given geometric mean condition and trigonometric identities, we can express

\cos 2\beta

in terms of

\alpha

and simplify it to the given form.

42 Medium

42 of 52

Application +4 | -1

The value of

\cos A \cos 2 A \cos 2^2 A \cos 2^3 A \ldots \cos 2^{n-1} A

View answer and explanation

Correct option: B

\frac{\sin 2^n A}{2^n \sin A}

Option B is correct because the given expression can be simplified using the identity

2 \sin A \cos A = \sin 2A

repeatedly.

43 Medium

43 of 52

Analyzing +4 | -1

\cos A + \cos B + \cos C = 0

, then

\cos \,\,3A + \cos \,\,3B + \cos \,\,3C

is equal to

View answer and explanation

Correct option: B

12\,\,\cos A\,\,\cos B\,\,\cos C

Option B (

12\\,\cos A\\,\cos B\\,\cos C

) is the correct answer because it follows from trigonometric identities and the given condition

\cos A + \cos B + \cos C = 0

44 Medium

44 of 52

Application +4 | -1

\sum \limits_{r = 1}^{n - 1} \,\,{\cos ^2}\,\,\frac{{r\;\pi }}{n}

is equal to

View answer and explanation

Correct option: C

\frac{n}{2} - 1

Option C (

\frac{n}{2} - 1

) is correct because the summation of the cosine squared terms simplifies to this expression using trigonometric identities and series summation.

45 Medium

45 of 52

Application +4 | -1

The smallest values of

x

and

y

satisfying

x-y=\frac{\pi}{4}

and

\cot x+\cot y=2

are -

View answer and explanation

Correct option: B

x=\frac{5 \pi}{12}, y=\frac{\pi}{6}

Option B (

x=\frac{5 \pi}{12}, y=\frac{\pi}{6}

) is correct because these values satisfy both the given equations:

x-y=\frac{\pi}{4}

and

\cot x+\cot y=2

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46 Medium

46 of 52

Analyzing +4 | -1

a = \tan \,\,27\theta - \tan \,\,\theta

and

b = \frac{{\sin \,\,\theta }}{{\cos \,\,3\theta }} + \frac{{\sin \,\,3\theta }}{{\cos \,\,9\theta }} + \frac{{\sin \,\,9\theta }}{{\cos \,\,27\theta }}

, then

View answer and explanation

Correct option: B

a = 2b

Option B (

a = 2b

) is the correct answer because, after simplification and trigonometric manipulations, the expression for

a

can be shown to be twice the expression for

b

47 Medium

47 of 52

Analyzing +4 | -1

For any angle

\theta

, the expression

\frac{{2\cos \,\,8\theta + 1}}{{2\cos \,\,\theta + 1}}

is equal to

View answer and explanation

Correct option: C

\left( {2\,\,\cos \,\,\theta - 1} \right)\,\,\left( {2\,\,\cos \,\,2\theta - 1} \right)\,\,\left( {2\,\,\cos \,\,4\theta - 1} \right)

Option C (

\left( {2\,\cos \,\theta - 1} \right)\,\left( {2\,\cos \,2\theta - 1} \right)\,\left( {2\,\cos \,4\theta - 1} \right)

) is correct because it is obtained by repeatedly applying the double angle formula for cosine and simplifying the expression.

48 Medium

48 of 52

Application +4 | -1

4n\;\alpha = \pi

, then the value of

\tan \,\,\alpha \,\,\tan \,\,2\alpha \,\,\tan \,\,3\alpha \,\,\tan \,\,4\alpha \ldots \tan \left( {2n - 2} \right)\alpha \,\,\tan \left( {2n - 1} \right)\alpha

, is

View answer and explanation

Correct option: B

Option B is correct because the product telescopes to 1 due to the complementary angle relationship between tangent functions.

49 Medium

49 of 52

Application +4 | -1

\cos (\alpha+\beta)=0

then

\operatorname{cosec}(\alpha-\beta)=

View answer and explanation

Correct option: D

\pm \sec 2 \beta

Option D (

\pm \sec 2 \beta

) is correct because from

\cos(\alpha + \beta) = 0

, we can derive that

\alpha + \beta = (2n+1)\frac{\pi}{2}

, and then find

\operatorname{cosec}(\alpha - \beta)

in terms of

\beta

50 Medium

50 of 52

Analyzing +4 | -1

Value of

{\cos ^2}\left( {A - B} \right)\,\, + \,\,{\cos ^2}B\,\, - \,\,2\cos \left( {A - B} \right)\cos A\,\,\cos B

View answer and explanation

Correct option: B

{\sin ^2}A

Option B (

{\sin ^2}A

) is correct because by applying trigonometric identities and simplification, the expression reduces to

{\sin ^2}A

51 Medium

51 of 52

Application +4 | -1

3 \sin \theta+5 \cos \theta=5

, then the value of

5 \sin \theta-3 \cos \theta

is equal to

View answer and explanation

Correct option: B

Option B (3) is correct because by using trigonometric identities and the given equation, the value of

5 \sin \theta - 3 \cos \theta

can be derived to be 3.

52 Medium

52 of 52

Application +4 | -1

\cos ^2 x+\cos ^4 x=1

, then the value of

\tan ^4 x+\cot ^4 x+\tan ^2 x-\cot ^2 x

is equal to

View answer and explanation

Correct option: B

Option B (2) is correct because by using the given equation and trigonometric identities, the expression simplifies to 2.

Revision notes and key takeaways

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