Properties of Inverse Trigonometric Functions

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JEE Main 2015

Let $tan^{- 1} y = tan^{- 1} x + tan^{- 1} (\frac{2 x}{1 - x ^{2}})$ , where $∣ x ∣ < \frac{1}{3}$ . Then a value of $y$ is :

\frac{3 x - x ^{3}}{1 + 3 x ^{2}}

(B)

\frac{3 x + x ^{3}}{1 + 3 x ^{2}}

(C)

\frac{3 x - x ^{3}}{1 - 3 x ^{2}}

(D)

\frac{3 x + x ^{3}}{1 - 3 x ^{2}}

JEE Main 2013

LEVELJEE Main

The value of $cot (cosec^{- 1} \frac{5}{3} + tan^{- 1} \frac{2}{3})$ is

If $x, y, z$ are in A.P. and $tan^{- 1} x, tan^{- 1} y$ and $tan^{- 1} z$ are also in A.P., then

Let $f : (- 1, 1) \to B$ , be a function defined by $f (x) = tan^{- 1} (\frac{2 x}{1 - x ^{2}})$ , then $f$ is both one - one and onto when $B$ is the interval

(A)

(0, π /2)

(B)

[0, π /2)

(C)

(- π /2, π /2)

(D)

(- π /2, π /2]

JEE Main 2005

LEVELJEE Main

If $cos^{- 1} x - cos^{- 1} \frac{y}{2} = α$ , then $4 x^{2} - 4 x y cos α + y^{2}$ is equal to

If $α = 3 sin^{- 1} (\frac{6}{11})$ and $β = 3 cos^{- 1} (\frac{4}{9})$ , where the inverse trigonometric functions take only the principal values, then the correct option(s) is (are)

* Multiple Correct Options

(A)

cos β > 0

(B)

sin β < 0

(C)

cos (α + β) > 0

(D)

cos α < 0

JEE Advanced 2013

LEVELJEE Main

The value of $cot (\sum_{n = 1}^{23} cot^{- 1} (1 + \sum_{k = 1}^{n} 2 k))$ is

$1 + x^{2} [{x cos (cot^{- 1} x) + sin (cot^{- 1} x)}^{2} - 1]^{1/2} =$

(A)

\frac{x}{1 + x ^{2}}

(B)

(C)

1 + x^{2}

(D)

1 + x^{2}

JEE Advanced 2001

LEVELJEE Main

If $sin^{- 1} (x - \frac{x ^{2}}{2} + \frac{x ^{3}}{4} - \dots) + cos^{- 1} (x^{2} - \frac{x ^{4}}{2} + \frac{x ^{6}}{4} - \dots) = \frac{π}{2}$ for $0 < ∣ x ∣ < 2$ , then $x$ equals

If we consider only the principle values of the inverse trigonometric functions then the value of $tan (cos^{- 1} \frac{1}{5 2} - sin^{- 1} \frac{4}{17})$ is

(A)

\frac{29}{3}

(B)

\frac{29}{3}

(C)

\frac{3}{29}

(D)

\frac{3}{29}

JEE Advanced 1986

LEVELJEE Main

The principal value of $sin^{- 1} (sin \frac{2 π}{3})$ is

(A)

2\pi/3

(B)

π /3

The value of $tan [cos^{- 1} (\frac{4}{5}) + tan^{- 1} (\frac{2}{3})]$ is

Let $f : [0, 4 π] \to [0, π]$ be defined by $f (x) = cos^{- 1} (cos x)$ . The number of points $x \in [0, 4 π]$ satisfying the equation $f (x) = \frac{10 - x}{10}$ is

JEE Advanced 1981

LEVELJEE Main

Let $a, b, c$ be positive real numbers. Let $θ = tan^{- 1} \frac{a ( a + b + c )}{b c} + tan^{- 1} \frac{b ( a + b + c )}{c a} + tan^{- 1} \frac{c ( a + b + c )}{ab}$ . Then $tan θ =$ .........

JEE Advanced 1989

LEVELJEE Main

The greater of the two angles $A = 2 tan^{- 1} (22 - 1)$ and $B = 3 sin^{- 1} (1/3) + sin^{- 1} (3/5)$ is .........

JEE Advanced 1984

LEVELBoard

The numerical value of $tan {2 tan^{- 1} (\frac{1}{5}) - \frac{π}{4}}$ is equal to .........

JEE Advanced 1981

LEVELJEE Main

Find the value of : $cos (2 cos^{- 1} x + sin^{- 1} x)$ at $x = \frac{1}{5}$ , where $0 \leq cos^{- 1} x \leq π$ and $- π /2 \leq sin^{- 1} x \leq π /2$ .

JEE Advanced 2002

LEVELJEE Main

Prove that $cos tan^{- 1} sin cot^{- 1} x = \frac{x ^{2} + 1}{x ^{2} + 2}$ .

JEE Advanced 1993

LEVELJEE Advanced

Using mathematical induction, prove that $tan^{- 1} (1/3) + tan^{- 1} (1/7) + .... tan^{- 1} {1/ (n^{2} + n + 1)} = tan^{- 1} {n / (n + 2)}$

JEE Main 2018 (15 April Evening)

LEVELJEE Main

If $f (x) = sin^{- 1} (\frac{2 \cdot 3 ^{x}}{1 + 9 ^{x}})$ , then $f^{'} (- \frac{1}{2})$ equals :-

(A)

3 lo g_{e} 3

(B)

- 3 lo g_{e} 3

(C)

- 3 lo g_{e} 3

(D)

3 lo g_{e} 3

JEE Main 2019 (9 January)

LEVELBoard

If $cos^{- 1} (\frac{2}{3 x}) + cos^{- 1} (\frac{3}{4 x}) = \frac{π}{2} (x > \frac{3}{4})$ , then x is equal to :

(A)

\frac{145}{12}

(B)

\frac{145}{10}

(C)

\frac{146}{12}

(D)

\frac{145}{11}

JEE Main 2019 (9 January)

LEVELJEE Advanced

If $x = sin^{- 1} (sin 10)$ and $y = cos^{- 1} (cos 10)$ , then $y - x$ is equal to:

(A)

π

(B)

7 π

(C)

(D)

JEE Main 2019 (8 April Shift 1)

LEVELJEE Main

If $α = cos^{- 1} (\frac{3}{5}), β = tan^{- 1} (\frac{1}{3})$ , where $0 < α, β < \frac{π}{2}$ , then $α - β$ is equal to :

(A)

sin^{- 1} (\frac{9}{5 10})

(B)

tan^{- 1} (\frac{9}{14})

(C)

cos^{- 1} (\frac{9}{5 10})

(D)

tan^{- 1} (\frac{9}{5 10})

JEE Main 2019 (10 April Shift 2)

LEVELJEE Main

If $cos^{- 1} x - cos^{- 1} \frac{y}{2} = α$ , where $- 1 \leq x \leq 1, - 2 \leq y \leq 2, x \leq \frac{y}{2}$ , then for all $x, y, 4 x^{2} - 4 x y cos α + y^{2}$ is equal to

(A)

4 \sin^2 \alpha - 2x^2y^2

(B)

4 \cos^2 \alpha + 2x^2y^2

JEE Main 2019 (12 April Shift 1)

LEVELJEE Main

The value of $sin^{- 1} (\frac{12}{13}) - sin^{- 1} (\frac{3}{5})$ is equal to :

(A)

π - sin^{- 1} (\frac{63}{65})

(B)

π - cos^{- 1} (\frac{33}{65})

(C)

\frac{π}{2} - sin^{- 1} (\frac{56}{65})

(D)

\frac{π}{2} - cos^{- 1} (\frac{9}{65})

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.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The value of cot(cosec−135​+tan−132​) is

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.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let f:(−1,1)→B, be a function defined by f(x)=tan−1(1−x22x​), then f is both one - one and onto when B is the interval

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.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If α=3sin−1(116​) and β=3cos−1(94​), where the inverse trigonometric functions take only the principal values, then the correct option(s) is (are)

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The value of cot(∑n=123​cot−1(1+∑k=1n​2k)) is

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.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If sin−1(x−2x2​+4x3​−…)+cos−1(x2−2x4​+4x6​−…)=2π​ for 0<∣x∣<2​, then x equals

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If we consider only the principle values of the inverse trigonometric functions then the value of tan(cos−152​1​−sin−117​4​) is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The principal value of sin−1(sin32π​) is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The value of tan[cos−1(54​)+tan−1(32​)] is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let f:[0,4π]→[0,π] be defined by f(x)=cos−1(cosx). The number of points x∈[0,4π] satisfying the equation f(x)=1010−x​ is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let a,b,c be positive real numbers. Let θ=tan−1bca(a+b+c)​​+tan−1cab(a+b+c)​​+tan−1abc(a+b+c)​​. Then tanθ= .........

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The greater of the two angles A=2tan−1(22​−1) and B=3sin−1(1/3)+sin−1(3/5) is .........

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The numerical value of tan{2tan−1(51​)−4π​} is equal to .........

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Find the value of : cos(2cos−1x+sin−1x) at x=51​, where 0≤cos−1x≤π and −π/2≤sin−1x≤π/2.

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Prove that costan−1sincot−1x=x2+2x2+1​​.

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Using mathematical induction, prove that tan−1(1/3)+tan−1(1/7)+....tan−1{1/(n2+n+1)}=tan−1{n/(n+2)}

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.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If cos−1(3x2​)+cos−1(4x3​)=2π​(x>43​), then x is equal to :

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If x=sin−1(sin10) and y=cos−1(cos10), then y−x is equal to:

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If α=cos−1(53​),β=tan−1(31​), where 0<α,β<2π​, then α−β is equal to :

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.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The value of sin−1(1312​)−sin−1(53​) is equal to :

Let $tan^{- 1} y = tan^{- 1} x + tan^{- 1} (\frac{2 x}{1 - x ^{2}})$ , where $∣ x ∣ < \frac{1}{3}$ . Then a value of $y$ is :

The value of $cot (cosec^{- 1} \frac{5}{3} + tan^{- 1} \frac{2}{3})$ is

If $x, y, z$ are in A.P. and $tan^{- 1} x, tan^{- 1} y$ and $tan^{- 1} z$ are also in A.P., then

Let $f : (- 1, 1) \to B$ , be a function defined by $f (x) = tan^{- 1} (\frac{2 x}{1 - x ^{2}})$ , then $f$ is both one - one and onto when $B$ is the interval

If $cos^{- 1} x - cos^{- 1} \frac{y}{2} = α$ , then $4 x^{2} - 4 x y cos α + y^{2}$ is equal to

If $α = 3 sin^{- 1} (\frac{6}{11})$ and $β = 3 cos^{- 1} (\frac{4}{9})$ , where the inverse trigonometric functions take only the principal values, then the correct option(s) is (are)

The value of $cot (\sum_{n = 1}^{23} cot^{- 1} (1 + \sum_{k = 1}^{n} 2 k))$ is

$1 + x^{2} [{x cos (cot^{- 1} x) + sin (cot^{- 1} x)}^{2} - 1]^{1/2} =$

If $sin^{- 1} (x - \frac{x ^{2}}{2} + \frac{x ^{3}}{4} - \dots) + cos^{- 1} (x^{2} - \frac{x ^{4}}{2} + \frac{x ^{6}}{4} - \dots) = \frac{π}{2}$ for $0 < ∣ x ∣ < 2$ , then $x$ equals

If we consider only the principle values of the inverse trigonometric functions then the value of $tan (cos^{- 1} \frac{1}{5 2} - sin^{- 1} \frac{4}{17})$ is

The principal value of $sin^{- 1} (sin \frac{2 π}{3})$ is

The value of $tan [cos^{- 1} (\frac{4}{5}) + tan^{- 1} (\frac{2}{3})]$ is

Let $f : [0, 4 π] \to [0, π]$ be defined by $f (x) = cos^{- 1} (cos x)$ . The number of points $x \in [0, 4 π]$ satisfying the equation $f (x) = \frac{10 - x}{10}$ is

Let $a, b, c$ be positive real numbers. Let $θ = tan^{- 1} \frac{a ( a + b + c )}{b c} + tan^{- 1} \frac{b ( a + b + c )}{c a} + tan^{- 1} \frac{c ( a + b + c )}{ab}$ . Then $tan θ =$ .........

The greater of the two angles $A = 2 tan^{- 1} (22 - 1)$ and $B = 3 sin^{- 1} (1/3) + sin^{- 1} (3/5)$ is .........

The numerical value of $tan {2 tan^{- 1} (\frac{1}{5}) - \frac{π}{4}}$ is equal to .........

Find the value of : $cos (2 cos^{- 1} x + sin^{- 1} x)$ at $x = \frac{1}{5}$ , where $0 \leq cos^{- 1} x \leq π$ and $- π /2 \leq sin^{- 1} x \leq π /2$ .

Prove that $cos tan^{- 1} sin cot^{- 1} x = \frac{x ^{2} + 1}{x ^{2} + 2}$ .

Using mathematical induction, prove that $tan^{- 1} (1/3) + tan^{- 1} (1/7) + .... tan^{- 1} {1/ (n^{2} + n + 1)} = tan^{- 1} {n / (n + 2)}$

If $f (x) = sin^{- 1} (\frac{2 \cdot 3 ^{x}}{1 + 9 ^{x}})$ , then $f^{'} (- \frac{1}{2})$ equals :-

If $cos^{- 1} (\frac{2}{3 x}) + cos^{- 1} (\frac{3}{4 x}) = \frac{π}{2} (x > \frac{3}{4})$ , then x is equal to :

If $x = sin^{- 1} (sin 10)$ and $y = cos^{- 1} (cos 10)$ , then $y - x$ is equal to:

If $α = cos^{- 1} (\frac{3}{5}), β = tan^{- 1} (\frac{1}{3})$ , where $0 < α, β < \frac{π}{2}$ , then $α - β$ is equal to :

If $cos^{- 1} x - cos^{- 1} \frac{y}{2} = α$ , where $- 1 \leq x \leq 1, - 2 \leq y \leq 2, x \leq \frac{y}{2}$ , then for all $x, y, 4 x^{2} - 4 x y cos α + y^{2}$ is equal to

The value of $sin^{- 1} (\frac{12}{13}) - sin^{- 1} (\frac{3}{5})$ is equal to :