Multiple and Sub-multiple Angles

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JEE Advanced 2013

Let $f : (- 1, 1) \to R$ be such that $f (cos 4 θ) = \frac{2}{2 - s e c ^{2} θ}$ for $θ \in (0, π /4) \cup (π /4, π /2)$ . Then the value(s) of $f (1/3)$ is (are)

1 - 3/2

(B)

1 + 3/2

(C)

1 - 2/3

(D)

1 + 2/3

JEE Advanced 1998

LEVELJEE Main

Let $n$ be an odd integer. If $sin n θ = \sum_{r = 0}^{n} b_{r} sin^{r} θ$ , for every value of $θ$ , then

(A)

b_{0} = 1, b_{1} = 3

(B)

b_{0} = 0, b_{1} = n

(C)

b_{0} = - 1, b_{1} = n

(D)

b_{0} = 0, b_{1} = n^{2} - 3 n + 3

JEE Advanced 1994

LEVELJEE Main

Let $0 < x < π /4$ then $(sec 2 x - tan 2 x)$ equals

(A)

tan (x - π /4)

(B)

tan (π /4 - x)

(C)

tan (x + π /4)

(D)

tan^{2} (x + π /4)

JEE Advanced 1992

LEVELJEE Main

Show that the value of $\frac{t a n x}{t a n 3 x}$ , wherever defined never lies between $1/3$ and 3.

JEE Main 2017

LEVELJEE Main

If $5 (tan^{2} x - cos^{2} x) = 2 cos 2 x + 9$ , then the value of $cos 4 x$ is:

(A)

- \frac{3}{5}

(B)

\frac{1}{3}

(C)

\frac{2}{9}

(D)

- \frac{7}{9}

JEE Main 2019 (10 January)

LEVELJEE Main

The value of $cos \frac{π}{2 ^{2}} \cdot cos \frac{π}{2 ^{3}} \cdot ..... \cdot cos \frac{π}{2 ^{10}} \cdot sin \frac{π}{2 ^{10}}$ is :

(A)

\frac{1}{256}

(B)

\frac{1}{2}

(C)

\frac{1}{512}

(D)

\frac{1}{1024}

JEE Main 2020 (9 January Shift 1)

LEVELJEE Main

The value of $cos^{3} \frac{π}{8} cos \frac{3 π}{8} + sin^{3} \frac{π}{8} sin \frac{3 π}{8}$ is:

(A)

\frac{1}{4}

(B)

\frac{1}{2 2}

(C)

\frac{1}{2}

(D)

\frac{1}{2}

JEE Main 2020 (9 Jan Morning)

LEVELJEE Main

Value of $cos^{3} \frac{π}{8} cos \frac{3 π}{8} + sin^{3} \frac{π}{8} sin \frac{3 π}{8}$ is

(A)

\frac{1}{2 2}

(B)

\frac{1}{2}

(C)

\frac{1}{2}

(D)

- \frac{1}{2}

JEE Main 2021 (27 July Shift 1)

LEVELBoard

If $sin θ + cos θ = \frac{1}{2}$ , then $16 (sin (2 θ) + cos (4 θ) + sin (6 θ))$ is equal to:

JEE Main 2022 (27 June Shift 2)

LEVELJEE Main

$α = sin 3 6^{\circ}$ is a root of which of the following equation

(A)

10 x^{4} - 10 x^{2} - 5 = 0

(B)

16 x^{4} + 20 x^{2} - 5 = 0

(C)

16 x^{4} - 20 x^{2} + 5 = 0

(D)

16 x^{4} - 10 x^{2} + 5 = 0

JEE Main 2023 (08 Apr Shift 2)

LEVELJEE Main

The value of $36 (4 cos^{2} 9^{\circ} - 1) (4 cos^{2} 2 7^{\circ} - 1) (4 cos^{2} 8 1^{\circ} - 1) (4 cos^{2} 24 3^{\circ} - 1)$ is

JEE Main 2026 (23 January Shift 2)

LEVELJEE Main

The least value of $(cos^{2} θ - 6 sin θ cos θ + 3 sin^{2} θ + 2)$ is

Let $G$ be a circle of radius $R > 0$ . Let $G_{1}, G_{2}, \dots, G_{n}$ be $n$ circles of equal radius $r > 0$ . Suppose each of the $n$ circles $G_{1}, G_{2}, \dots, G_{n}$ touches the circle $G$ externally. Also, for $i = 1, 2, \dots, n - 1$ , the circle $G_{i}$ touches $G_{i + 1}$ externally, and $G_{n}$ touches $G_{1}$ externally. Then, which of the following statements is/are TRUE?

* Multiple Correct Options

(A)

n = 4

, then

(2 - 1) r < R

(B)

n = 5

, then

r < R

(C)

n = 8

, then

(2 - 1) r < R

(D)

n = 12

, then

2 (3 + 1) r > R

JEE Advanced 2024

LEVELJEE Main

Let $\frac{π}{2} < x < π$ be such that $cot x = - \frac{5}{11}$ . Then $sin (\frac{11 x}{2}) (sin 6 x - cos 6 x) + cos (\frac{11 x}{2}) (sin 6 x + cos 6 x)$ is equal to

(A)

\frac{11 - 1}{2 3}

(B)

\frac{11 + 1}{2 3}

(C)

\frac{11 + 1}{3 2}

(D)

\frac{11 - 1}{3 2}

JEE(ADVANCED)-201

LEVELJEE Main

Let $α$ and $β$ be nonzero real numbers such that $2 (cos β - cos α) + cos α cos β = 1$ . Then which of the following is/are true?

* Multiple Correct Options

(A)

tan (\frac{α}{2}) + 3 tan (\frac{β}{2}) = 0

(B)

3 tan (\frac{α}{2}) + tan (\frac{β}{2}) = 0

(C)

tan (\frac{α}{2}) - 3 tan (\frac{β}{2}) = 0

(D)

3 tan (\frac{α}{2}) - tan (\frac{β}{2}) = 0

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let f:(−1,1)→R be such that f(cos4θ)=2−sec2θ2​ for θ∈(0,π/4)∪(π/4,π/2). Then the value(s) of f(1/3) is (are)

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let n be an odd integer. If sinnθ=∑r=0n​br​sinrθ, for every value of θ, then

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let 0<x<π/4 then (sec2x−tan2x) equals

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Show that the value of tan3xtanx​, wherever defined never lies between 1/3 and 3.

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If 5(tan2x−cos2x)=2cos2x+9, then the value of cos4x is:

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The value of cos22π​⋅cos23π​⋅.....⋅cos210π​⋅sin210π​ is :

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The value of cos38π​cos83π​+sin38π​sin83π​ is:

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Value of cos38π​cos83π​+sin38π​sin83π​ is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If sinθ+cosθ=21​, then 16(sin(2θ)+cos(4θ)+sin(6θ)) is equal to:

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } α=sin36∘ is a root of which of the following equation

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The value of 36(4cos29∘−1)(4cos227∘−1)(4cos281∘−1)(4cos2243∘−1) is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The least value of (cos2θ−6sinθcosθ+3sin2θ+2) is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let 2π​<x<π be such that cotx=−11​5​. Then sin(211x​)(sin6x−cos6x)+cos(211x​)(sin6x+cos6x) is equal to

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let α and β be nonzero real numbers such that 2(cosβ−cosα)+cosαcosβ=1. Then which of the following is/are true?

Let $f : (- 1, 1) \to R$ be such that $f (cos 4 θ) = \frac{2}{2 - s e c ^{2} θ}$ for $θ \in (0, π /4) \cup (π /4, π /2)$ . Then the value(s) of $f (1/3)$ is (are)

Let $n$ be an odd integer. If $sin n θ = \sum_{r = 0}^{n} b_{r} sin^{r} θ$ , for every value of $θ$ , then

Let $0 < x < π /4$ then $(sec 2 x - tan 2 x)$ equals

Show that the value of $\frac{t a n x}{t a n 3 x}$ , wherever defined never lies between $1/3$ and 3.

If $5 (tan^{2} x - cos^{2} x) = 2 cos 2 x + 9$ , then the value of $cos 4 x$ is:

The value of $cos \frac{π}{2 ^{2}} \cdot cos \frac{π}{2 ^{3}} \cdot ..... \cdot cos \frac{π}{2 ^{10}} \cdot sin \frac{π}{2 ^{10}}$ is :

The value of $cos^{3} \frac{π}{8} cos \frac{3 π}{8} + sin^{3} \frac{π}{8} sin \frac{3 π}{8}$ is:

Value of $cos^{3} \frac{π}{8} cos \frac{3 π}{8} + sin^{3} \frac{π}{8} sin \frac{3 π}{8}$ is

If $sin θ + cos θ = \frac{1}{2}$ , then $16 (sin (2 θ) + cos (4 θ) + sin (6 θ))$ is equal to:

$α = sin 3 6^{\circ}$ is a root of which of the following equation

The value of $36 (4 cos^{2} 9^{\circ} - 1) (4 cos^{2} 2 7^{\circ} - 1) (4 cos^{2} 8 1^{\circ} - 1) (4 cos^{2} 24 3^{\circ} - 1)$ is

The least value of $(cos^{2} θ - 6 sin θ cos θ + 3 sin^{2} θ + 2)$ is

Let $\frac{π}{2} < x < π$ be such that $cot x = - \frac{5}{11}$ . Then $sin (\frac{11 x}{2}) (sin 6 x - cos 6 x) + cos (\frac{11 x}{2}) (sin 6 x + cos 6 x)$ is equal to

Let $α$ and $β$ be nonzero real numbers such that $2 (cos β - cos α) + cos α cos β = 1$ . Then which of the following is/are true?