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JEE • MATHEMATICS • INVERSE TRIGONOMETRY

Inverse trigonometry practice · Medium

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Practice format and chapter quiz

These sets use single-correct-answer MCQs. They do not cover every JEE Main or JEE Advanced question format and are not full-length mock papers.

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Showing 1-30 of 221 questions
1 Medium
1 of 221
Application +4 | -1
The value of cos⁡−1(cos⁡12)−sin⁡−1(sin⁡12)\cos ^{-1}(\cos 12)-\sin ^{-1}(\sin 12) is-
View answer and explanation
Correct option: C
8π−48 \pi-4
Option C (8π−248 \pi - 24) is correct because it involves finding the correct range for the inverse trigonometric functions and then subtracting them.
2 Medium
2 of 221
Application +4 | -1
The value of sin⁡(2tan⁡−1(1/3))+\sin \left(2 \tan ^{-1}(1 / 3)\right)+ cos⁡(tan⁡−122)\cos \left(\tan ^{-1} 2 \sqrt{2}\right) is
View answer and explanation
Correct option: C
14/1514 / 15
Option C (14/15) is correct because by evaluating the sine and cosine terms using trigonometric identities and substituting, we arrive at the correct answer.
3 Medium
3 of 221
Application +4 | -1
Find cos⁡−1{12(cos⁡9π10−sin⁡9π10)}=\cos ^ {-1}\left\{\frac{1}{\sqrt{2}}\left(\cos \frac{9 \pi}{10}-\sin \frac{9 \pi}{10}\right)\right\}=
View answer and explanation
Correct option: D
17π20\frac{17 \pi}{20}
Option D (17π20\frac{{17 \pi}}{{20}}) is correct because it simplifies the given expression using trigonometric identities and inverse cosine properties.
4 Medium
4 of 221
Application +4 | -1
The number of positive integral solutions of the equation tan⁡−1x+cot⁡−1y=tan⁡−14\tan ^{-1} x+\cot ^{-1} y=\tan ^{-1} 4 is
View answer and explanation
Correct option: B
1
Option B (1) is correct because by simplifying the equation, we find only one positive integral solution that satisfies the given condition.
5 Medium
5 of 221
Application +4 | -1
If sin⁡−12a1+a2−cos⁡−11−b21+b2=tan⁡−12x1−x2{\sin ^{ - 1}}\frac{{2a}}{{1 + {a^2}}} - {\cos ^{ - 1}}\frac{{1 - {b^2}}}{{1 + {b^2}}} = {\tan ^{ - 1}}\frac{{2x}}{{1 - {x^2}}}, then the value of xx is
View answer and explanation
Correct option: D
a−b1+ab\frac{{a - b}}{{1 + ab}}
Option D (a−b1+ab\frac{{a - b}}{{1 + ab}}) is correct because it is derived by using trigonometric identities to simplify the given equation and solve for xx.
6 Medium
6 of 221
Application +4 | -1
The value of α22cosec⁡2(12tan⁡−1αβ)−β22sec⁡2(12tan⁡−1βα)\frac{\alpha^2}{2} \operatorname{cosec}^2\left(\frac{1}{2} \tan ^{-1} \frac{\alpha}{\beta}\right)-\frac{\beta^2}{2} \sec ^2\left(\frac{1}{2} \tan ^{-1} \frac{\beta}{\alpha}\right) is equal to
View answer and explanation
Correct option: D
α2+β2(β+α)\sqrt{\alpha^2+\beta^2}(\beta+\alpha)
Option D (α2+β2(β+α)\sqrt{{\alpha^2+\beta^2}}(\beta+\alpha)) is correct because after simplifying the trigonometric expressions and substituting, the value equals to the given expression.
7 Medium
7 of 221
Application +4 | -1
2(tan⁡−11+tan⁡−12+tan⁡−13)2\left(\tan ^{-1} 1+\tan ^{-1} 2+\tan ^{-1} 3\right) is equal to
View answer and explanation
Correct option: D
2π2 \pi
Option D (2π2 \pi) is correct because the expression 2(tan⁡−11+tan⁡−12+tan⁡−13)2(\tan^{-1} 1 + \tan^{-1} 2 + \tan^{-1} 3) simplifies to 2π2\pi using the properties of inverse tangent functions.
8 Medium
8 of 221
Application +4 | -1
sin⁡−1x5+cosec⁡−154=π2\sin ^{-1} \frac{x}{5}+\operatorname{cosec}^{-1} \frac{5}{4}=\frac{\pi}{2} then x=x=
View answer and explanation
Correct option: D
3
Option D (3) is correct because using the identity sin⁡−1x+cos⁡−1x=π2\sin^{-1}x + \cos^{-1}x = \frac{\pi}{2} and cosec⁡−1x=sin⁡−11x\operatorname{cosec}^{-1} x = \sin^{-1} \frac{1}{x}, we can solve for x.
9 Medium
9 of 221
Application +4 | -1
Two angles of a triangle are sin⁡−1(1/5)\sin ^{-1}(1 / \sqrt{5}) and sin⁡−1(1/10)\sin ^{-1}(1 / \sqrt{10}) then the third angle is
View answer and explanation
Correct option: B
3π/43 \pi / 4
Option B, 3π/43 \pi / 4, is correct because by calculating the sine of the sum of the two given angles, we can find the third angle of the triangle.
10 Medium
10 of 221
Application +4 | -1
If cos⁡−1(23x)+cos⁡−1(34x)=π2(x>34)\cos ^{-1}\left(\frac{2}{3 x}\right)+\cos ^{-1}\left(\frac{3}{4 x}\right)=\frac{\pi}{2}\left(x>\frac{3}{4}\right),then xx is equal to:
View answer and explanation
Correct option: A
14512\frac{\sqrt{145}}{12}
The correct answer is A (14512\frac{\sqrt{145}}{12}) because it involves using inverse cosine properties and solving the resulting equation.
11 Medium
11 of 221
Application +4 | -1
If in a triangle, two angles are tan⁡−12\tan ^{-1} 2 and tan⁡−13\tan ^{-1} 3, then the third angle is
View answer and explanation
Correct option: B
3π4\frac{3 \pi}{4}
Option B (3π4\frac{{3 \pi}}{{4}}) is correct because the sum of the two given angles is greater than π2\frac{{\pi}}{{2}}, requiring the use of the adjusted tangent addition formula, leading to this third angle.
12 Medium
12 of 221
Application +4 | -1
If 2 tan⁡−1(cos  x)=tan⁡−1(2  cosec  x),  {\tan ^{ - 1}}\left( {\mathrm{cos}\;x} \right) = {\tan ^{ - 1}}\left( {2\;\mathrm{cosec}\;x} \right),\; then the value of xx is
View answer and explanation
Correct option: B
π4\frac{\pi }{4}
Option B is correct because solving the equation 2tan⁡−1(cos⁡x)=tan⁡−1(2cosec  x)2{\tan ^{ - 1}}(\cos x) = {\tan ^{ - 1}}(2\mathrm{cosec}\;x) leads to x=π4x = \frac{\pi }{4}.
13 Medium
13 of 221
Application +4 | -1
If sin⁡−1(15)+sec⁡−1(2)+2tan⁡−1(13)+sec⁡−1(5)\sin ^{-1}\left(\frac{1}{5}\right)+\sec ^{-1}(2)+2 \tan ^{-1}\left(\frac{1}{\sqrt{3}}\right)+\sec ^{-1}(5) +sin⁡−1(12)+2tan⁡−1(3)=kπ+\sin ^{-1}\left(\frac{1}{2}\right)+2 \tan ^{-1}(\sqrt{3})={k} \pi, then k={k}=
View answer and explanation
Correct option: A
2
Option A (2) is correct because by using trigonometric identities and simplifying, the value of k is found to be 2.
14 Medium
14 of 221
Application +4 | -1
If tan⁡−1x+tan⁡−1y+tan⁡−1z=π\tan ^{-1} x+\tan ^{-1} y+\tan ^{-1} z=\pi then 1xy+1yz+1zx\frac{1}{x y}+\frac{1}{y z}+\frac{1}{z x} is
View answer and explanation
Correct option: B
1
Option B (1) is correct because if tan⁡−1x+tan⁡−1y+tan⁡−1z=π\tan^{-1} x + \tan^{-1} y + \tan^{-1} z = \pi, then x+y+z=xyzx + y + z = xyz, which implies 1xy+1yz+1zx=1\frac{1}{xy} + \frac{1}{yz} + \frac{1}{zx} = 1.
15 Medium
15 of 221
Application +4 | -1
If tan⁡−1x−tan⁡−1y=tan⁡−1A\tan ^{-1} x-\tan ^{-1} y=\tan ^{-1} A, and xy>−1x y>-1, then AA is
View answer and explanation
Correct option: C
x−y1+xy\frac{x-y}{1+x y}
Option C (x−y1+xy\frac{x-y}{1+xy}) is correct because using the tangent subtraction formula, tan⁡−1x−tan⁡−1y=tan⁡−1x−y1+xy\tan^{-1} x - \tan^{-1} y = \tan^{-1} \frac{x-y}{1+xy}, which directly gives the value of A.

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16 Medium
16 of 221
Application +4 | -1
The value of cot⁡−11−sin⁡x+1+sin⁡x1−sin⁡x−1+sin⁡x\cot -1 \frac{\sqrt{1-\sin x}+\sqrt{1+\sin x}}{\sqrt{1-\sin x}-\sqrt{1+\sin x}} if 0∘<x<90∘0^{\circ} < x < 90^{\circ} is
View answer and explanation
Correct option: C
π−x2\pi-\frac{x}{2}
Option C (π−x2\pi - \frac{x}{2}) is correct because simplifying the expression inside the cot⁡−1\cot^{-1} function leads to −cot⁡(x2)-\cot(\frac{x}{2}), and using the property cot⁡−1(−x)=π−cot⁡−1(x)\cot^{-1}(-x) = \pi - \cot^{-1}(x) gives the result.
17 Medium
17 of 221
Application +4 | -1
If (tan⁡−1x)2+(cot⁡−1x)2=5π2/8\left(\tan ^{-1} x\right)^2+\left(\cot ^{-1} x\right)^2=5 \pi^2 / 8, then xx equals
View answer and explanation
Correct option: C
-1
Option C (-1) is correct because substituting x = -1 into the equation satisfies the given condition.
18 Medium
18 of 221
Application +4 | -1
cos⁡−1[cos⁡{2cot⁡−1(2−1)}]\cos ^{-1}\left[\cos \left\{2 \cot ^{-1}(\sqrt{2}-1)\right\}\right] is equal to
View answer and explanation
Correct option: D
3π4\frac{3 \pi}{4}
Option B (π212\frac{{\pi^2}}{{12}}) is correct because the integral is solved using properties of even functions and trigonometric substitutions.
19 Medium
19 of 221
Analyzing +4 | -1
If x1,x2,  x3,  x4{x_1},{x_2},\;{x_3},\;{x_4} are the roots of the equation x4−x3sin⁡2β−xcos⁡β−sin⁡β=0,  {x^4} - {x^3}\sin 2\mathrm{\beta } - x\cos \mathrm{\beta } - \sin \mathrm{\beta } = 0,\; then tan⁡−1+tan⁡−1x2+tan⁡−1x3+tan⁡−1x4{\tan ^{ - 1}} + {\tan ^{ - 1}}{x_2} + {\tan ^{ - 1}}{x_3} + {\tan ^{ - 1}}{x_4} is equal to
View answer and explanation
Correct option: B
π2−β\frac{\pi }{2} - \mathrm{\beta }
Option D (xx) is correct because simplifying the given expression using trigonometric identities leads to the result xx.
20 Medium
20 of 221
Application +4 | -1
Number of solutions to the equation sin⁡−1x−cos⁡−1x=cos⁡−1(32)\sin ^{-1} x-\cos ^{-1} x=\cos ^{-1}\left(\frac{\sqrt{3}}{2}\right) is
View answer and explanation
Correct option: A
One
Option A (One) is correct because solving the trigonometric equation yields only one valid solution for x.
21 Medium
21 of 221
Application +4 | -1
[tan⁡−1x]2−2[tan⁡−1x]+1≤0\left[\tan ^{-1} x\right]^2-2\left[\tan ^{-1} x\right]+1 \leq 0, where [x][x] denotes greatest interger ≤x\leq x, iff x∈[tan⁡k,tan⁡2)x \in[\tan k, \tan 2) then evaluate kk.
View answer and explanation
Correct option: A
1
Option A (1) is correct because solving the inequality leads to the condition that x belongs to the interval [tan 1, tan 2), thus k = 1.
22 Medium
22 of 221
Application +4 | -1
If sin⁡−1x+sin⁡−1y+sin⁡−1z=3π/2\sin ^{-1} x+\sin ^{-1} y+\sin ^{-1} z=3 \pi / 2, then the value of x100+y100+z100−9x101+y101+z101x^{100}+y^{100}+z^{100}-\frac{9}{x^{101}+y^{101}+z^{101}} is
View answer and explanation
Correct option: B
0
Option B (0) is correct because, given the condition sin⁡−1x+sin⁡−1y+sin⁡−1z=3π/2\sin^{-1}x + \sin^{-1}y + \sin^{-1}z = 3\pi/2, it implies x=y=z=1, which simplifies the expression to 0.
23 Medium
23 of 221
Analyzing +4 | -1
If sin⁡−1(x2−x42+x64……∞)+cos⁡−1(x−x22+x34….∞)=π2\sin ^{-1}\left(x^2-\frac{x^4}{2}+\frac{x^6}{4} \ldots \ldots \infty\right)+\cos ^{-1}\left(x-\frac{x^2}{2}+\frac{x^3}{4} \ldots . \infty\right)=\frac{\pi}{2} for 0<∣x∣<20 < |x| < \sqrt{2}, then cot⁡−1x\cot ^{-1} x equals to
View answer and explanation
Correct option: D
π4\frac{\pi}{4}
Option D (π4\frac{\pi}{4}) is correct because solving the given equation after simplifying the infinite geometric series leads to x=1x=1, and thus cot⁡−1(1)=π4\cot^{-1}(1) = \frac{\pi}{4}.
24 Medium
24 of 221
Application +4 | -1
If tan⁡−1a+tan⁡−1b=sin⁡−11−tan⁡−1c{\tan ^{ - 1}}a + {\tan ^{ - 1}}b = {\sin ^{ - 1}}1 - {\tan ^{ - 1}}c, then
View answer and explanation
Correct option: C
1a+1b+1c−1abc=0\frac{1}{a} + \frac{1}{b} + \frac{1}{c} - \frac{1}{{abc}} = 0
Option C (1a+1b+1c−1abc=0\frac{1}{a} + \frac{1}{b} + \frac{1}{c} - \frac{1}{{abc}} = 0) is correct because it is the result of simplifying the given equation using the tangent addition formula and the fact that sin⁡−11=π2{\sin ^{ - 1}}1 = \frac{\pi}{2}.
25 Medium
25 of 221
Application +4 | -1
If tan⁡−1a+xa+tan⁡−1a−xa=π6\tan ^{-1} \frac{a+x}{a}+\tan ^{-1} \frac{a-x}{a}=\frac{\pi}{6}, then x2x^2 is equal to
View answer and explanation
Correct option: C
23a22 \sqrt{3} a^2
Option C (23a22 \sqrt{3} a^2) is correct because it is the solution obtained after simplifying the given trigonometric equation using the tangent addition formula.
26 Medium
26 of 221
Application +4 | -1
If xr=cos⁡π2r+isin⁡π2rx_r=\cos \frac{\pi}{2^r}+i \sin \frac{\pi}{2^r}, then x1x2x3…∞=x_1 x_2 x_3 \ldots \infty=
View answer and explanation
Correct option: D
-1
Option D is correct because the infinite product simplifies to -1 using Euler's formula and properties of geometric series.
27 Medium
27 of 221
Application +4 | -1
If sin⁡−1x+sin⁡−1y+sin⁡−1z=3π2\sin ^{-1} x+\sin ^{-1} y+\sin ^{-1} z=\frac{3 \pi}{2} then the value of x100+y100+z100−3x101+y101+z101{x}^{100}+{y}^{100}+{z}^{100}-\frac{3}{{x}^{101}+{y}^{101}+{z}^{101}} is
View answer and explanation
Correct option: C
2
Option C (2) is correct because the value of the expression is 2, derived from the given condition on inverse sines.
28 Medium
28 of 221
Analyzing +4 | -1
The value of cot⁡(∑n=119cot⁡−1(1+∑p=1n2p))\cot \left(\sum_{n=1}^{19} \cot ^{-1}\left(1+\sum_{p=1}^{n} 2 p\right)\right) is:
View answer and explanation
Correct option: A
2119\frac{21}{19}
Option A (2119\frac{{21}}{{19}}) is correct because by simplifying the summation and using the arctangent subtraction formula, the expression reduces to 2119\frac{{21}}{{19}}.
29 Medium
29 of 221
Application +4 | -1
If asin⁡−1x−bcos⁡−1x=ca \sin ^{-1} x-b \cos ^{-1} x=c, then asin⁡−1x+a \sin ^{-1} x+ bcos⁡−1xb \cos ^{-1} x is equal to
View answer and explanation
Correct option: D
πab+c(a−b)a+b\frac{\pi a b+c(a-b)}{a+b}
Option D (πab+c(a−b)a+b\frac{\pi a b+c(a-b)}{a+b}) is correct because by solving the system of equations, we can find the value of the expression.
30 Medium
30 of 221
Application +4 | -1
If two angles of a Δ\Delta are tan⁡−112\tan ^{-1} \frac{1}{2} and tan⁡−11/3\tan ^{-1} 1 / 3, then find θ15\frac{\theta}{15} (where θ\theta is third angle of Δ\Delta in degree).
View answer and explanation
Correct option: A
9
Option A (9) is correct because the third angle is 135 degrees, and 135/15 = 9.
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