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JEE • MATHEMATICS • INVERSE TRIGONOMETRY

Inverse trigonometry practice · Mixed difficulty

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481 Easy
481 of 605
Application +4 | -1
θ=tan⁡−1(2tan⁡2θ)−tan⁡−1((1/3)tan⁡θ)\theta=\tan ^{-1}\left(2 \tan ^2 \theta\right)-\tan ^{-1}((1 / 3) \tan \theta) if tan⁡θ\tan \theta is equal to
View answer and explanation
Correct option: A
-2
Option A (-2) is correct because, after simplifying the equation using the tangent subtraction formula, we find that tan(theta) = -2 is a valid solution.
482 Easy
482 of 605
Application +4 | -1
The value of sin⁡−1(cos⁡(cos⁡−1(cos⁡x)+sin⁡−1(sin⁡x)))\sin ^{-1}\left(\cos \left(\cos ^{-1}(\cos x)+\sin ^{-1}(\sin x)\right)\right) where x∈(π2,π)x \in\left(\frac{\pi}{2}, \pi\right) is equal to
View answer and explanation
Correct option: A
−π/2-\pi / 2
Option A (−π/2-\pi / 2) is correct because the expression simplifies to sin⁡−1(cos⁡(π))=sin⁡−1(−1)=−π/2\sin^{-1}(\cos(\pi)) = \sin^{-1}(-1) = -\pi/2.
483 Easy
483 of 605
Application +4 | -1
2π−(sin⁡−145+sin⁡−1513+sin⁡−11665)2\pi - \left( {{{\sin }^{ - 1}}\frac{4}{5} + {{\sin }^{ - 1}}\frac{5}{{13}} + {{\sin }^{ - 1}}\frac{{16}}{{65}}} \right) is equal to
View answer and explanation
Correct option: C
3π2\frac{{3\pi }}{2}
Option C (3π2\frac{{3\pi }}{2}) is correct because it is the result of simplifying the given expression using trigonometric identities and inverse trigonometric functions.
484 Easy
484 of 605
Application +4 | -1
tan⁡−1xy−tan⁡−1x−yx+y  {\tan ^{ - 1}}\frac{x}{y} - {\tan ^{ - 1}}\frac{{x - y}}{{x + y}}\; is equal to

(where x<y>0)x < y > 0)
View answer and explanation
Correct option: B
π4\frac{\pi }{4}
Option B (π4\frac{\pi }{4}) is correct because by simplifying the expression using the tangent subtraction formula and the identity tan⁡−1(x)+cot⁡−1(x)=π2\tan^{-1}(x) + \cot^{-1}(x) = \frac{\pi}{2}, it simplifies to π4\frac{\pi}{4}.
485 Easy
485 of 605
Application +4 | -1
If x∈(−∞,  −1)x \in \left( { - \infty ,\; - 1} \right), then sin⁡−1(2x1+x2){\sin ^{ - 1}}\left( {\frac{{2x}}{{1 + {x^2}}}} \right) equals
View answer and explanation
Correct option: C
−π−2tan⁡−1x- \pi - 2{\tan ^{ - 1}}x
Option C (−π−2tan⁡−1x- \pi - 2{\tan ^{ - 1}}x) is correct because, within the specified range for xx, the expression simplifies to this form using trigonometric identities and considering the range of the arctangent function.
486 Easy
486 of 605
Application +4 | -1
The minimum value of 12π(cosec⁡−1[3x2+54]+sec⁡−1[3x2+14])\frac{12}{\pi}\left(\operatorname{cosec}^{-1}\left[3 x^2+\frac{5}{4}\right]+\sec ^{-1}\left[3 x^2+\frac{1}{4}\right]\right) is equal to (where [.] denotes the greatest integer function)
View answer and explanation
Correct option: A
2
Option A (2) is correct because by analyzing the range of the inverse trigonometric functions and the greatest integer function, the minimum value is found to be 2.
487 Medium
487 of 605
Application +4 | -1
The value of cos⁡−1(−1/2)−2sin⁡−1(1/2)+\cos ^{-1}(-1 / 2)-2 \sin ^{-1}(1 / 2)+ 3cos⁡−1(−1/2)−4tan⁡−1(−1)3 \cos ^{-1}(-1 / \sqrt{2})-4 \tan ^{-1}(-1) is equal to
View answer and explanation
Correct option: D
25π/1225 \pi / 12
Option D (25π/1225 \pi / 12) is correct because the expression evaluates to (2π/3)−2×(π/6)+3×(3π/4)−4(−π/4)=25π/12(2 \pi / 3)-2 \times (\pi / 6)+3 \times (3 \pi / 4)-4(-\pi / 4)=25 \pi/12.
488 Medium
488 of 605
Application +4 | -1
Let (sin⁡a)x2+(sin⁡a)x+(1−cos⁡a)=0(\sin a) x^2+(\sin a) x+(1-\cos a)=0. The value of a for which roots of this equation are real and distinct, is
View answer and explanation
Correct option: A
(0,2tan⁡−11/4]\left(0,2 \tan ^{-1} 1 / 4\right]
Option A ((0,2tan⁡−11/4]\left(0,2 \tan ^{{-1}} 1 / 4\right]) is correct because it represents the range of 'a' for which the discriminant of the quadratic equation is positive, ensuring real and distinct roots.
489 Easy
489 of 605
Application +4 | -1
If sin⁡−113+sin⁡−123=sin⁡−1x{\sin ^{ - 1}}\frac{1}{3} + {\sin ^{ - 1}}\frac{2}{3} = {\sin ^{ - 1}}x, then the value of xx is
View answer and explanation
Correct option: C
(5+42)9\frac{{\left( {\sqrt 5 + 4\sqrt 2 } \right)}}{9}
Option C ((5+42)9\frac{{\left( {\sqrt 5 + 4\sqrt 2 } \right)}}{9}) is correct because it uses the sine addition formula to find the value of x.
490 Medium
490 of 605
Analyzing +4 | -1
Let I1=π24+2,  I2=(tan⁡−1(1e))2+2ee2+1,I3=(tan⁡−1e)2+2e2+1I_1=\frac{\pi^2}{4}+\sqrt{2}, \, \, I_2=\left(\tan ^{-1}\left(\frac{1}{e}\right)\right)^2+\frac{2 e}{\sqrt{e^2+1}}, I_3=\left(\tan ^{-1} e\right)^2+\frac{2}{\sqrt{e^2+1}}, then which of the following is true?
View answer and explanation
Correct option: B
I2<I1<I3{I}_2 < {I}_1 < {I}_3
Option B (I2<I1<I3{I}_2 < {I}_1 < {I}_3) is correct because by analyzing the function f(x)=(tan⁡−1x)2+2x2+1f(x)=\left(\tan ^{-1} x\right)^2+\frac{2}{\sqrt{x^2+1}} and its derivative, we can determine the correct inequality.
491 Easy
491 of 605
Application +4 | -1
cos⁡(2cos⁡−1x)+cos⁡(2sin⁡−1x)=0\cos \left(2 \cos ^{-1} x\right)+\cos \left(2 \sin ^{-1} x\right)=0 for
View answer and explanation
Correct option: D
−1≤x≤1-1 \leq x \leq 1
Option D (-1 ≤ x ≤ 1) is correct because the given equation simplifies to 0 for all x within the domain of both inverse cosine and inverse sine functions.
492 Easy
492 of 605
Application +4 | -1
If tan⁡−1x+tan⁡−1y+tan⁡−1z=π{\tan ^{ - 1}}x + {\tan ^{ - 1}}y + {\tan ^{ - 1}}z = \pi, then the value of x+y+zx + y + z is
View answer and explanation
Correct option: B
xyzxyz
Option B (xyzxyz) is correct because it is the simplified form of the given trigonometric equation after applying inverse trigonometric identities.
493 Medium
493 of 605
Analyzing +4 | -1
S=tan⁡−1(1n2+n+1)+tan⁡−1(1n2+3n+3)+…S=\tan ^{-1}\left(\frac{1}{n^{2}+n+1}\right)+\tan ^{-1}\left(\frac{1}{n^{2}+3 n+3}\right)+\ldots
+tan⁡−1(11+(n+19)(n+20))+\tan ^{-1}\left(\frac{1}{1+(n+19)(n+20)}\right), then tan⁡S\tan S is equal to :
View answer and explanation
Correct option: C
20n2+20n+1\frac{20}{n^{2}+20 n+1}
The correct answer is C (20n2+20n+1\frac{20}{n^{2}+20 n+1}). This is found by recognizing the telescoping nature of the series and using the arctangent subtraction formula.
494 Easy
494 of 605
Application +4 | -1
If x>1x > 1 and 2tan⁡−1x−sin⁡−1(7/25)=π2 \tan ^{-1} x-\sin ^{-1}(7 / 25)=\pi then the value of xx is
View answer and explanation
Correct option: A
7
The correct answer is A (7) because using the given condition and trigonometric identities, we can solve for x.
495 Easy
495 of 605
Application +4 | -1
Value of cos⁡[2cos⁡−1(45)]\cos \left[2 \cos ^{-1}\left(\frac{4}{5}\right)\right] equals
View answer and explanation
Correct option: B
725\frac{7}{25}
Option A (12\frac{{1}}{{\sqrt{{2}}}}) is correct because it is the value obtained after simplifying the expression using trigonometric identities and the properties of inverse trigonometric functions.

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496 Easy
496 of 605
Analyzing +4 | -1
∑m=1ntan⁡−1(2mm4+m2+2)  \sum \limits_{m = 1}^n {\tan ^{ - 1}}\left( {\frac{{2m}}{{{m^4} + {m^2} + 2}}} \right)\; is equal to
View answer and explanation
Correct option: A
tan⁡−1(n2+nn2+n+2){\tan ^{ - 1}}\left( {\frac{{{n^2} + n}}{{{n^2} + n + 2}}} \right)
Option A is correct because the given summation can be simplified using the arctangent subtraction formula and telescoping series, resulting in the expression for option A.
497 Easy
497 of 605
Application +4 | -1
The value of tan⁡{cos⁡−1(−27)−π2}\tan \left\{ {{{\cos }^{ - 1}}\left( { - \frac{2}{7}} \right) - \frac{\pi }{2}} \right\} is
View answer and explanation
Correct option: A
235\frac{2}{{3\sqrt 5 }}
Option A (235\frac{2}{{3\sqrt 5 }}) is correct because it is the simplified value obtained by using trigonometric identities and the definition of inverse trigonometric functions.
498 Easy
498 of 605
Application +4 | -1
Let tan⁡−1y=tan⁡−1x+tan⁡−1(2x1−x2),{\tan ^{ - 1}}y = {\tan ^{ - 1}}x + {\tan ^{ - 1}}\left( {\frac{{2x}}{{1 - {x^2}}}} \right), where ∣x∣<13.\left| x \right| < \frac{1}{{\sqrt 3 }}. Then, a value of yy is
View answer and explanation
Correct option: A
3x−x31−3x2\frac{{3x - {x^3}}}{{1 - 3{x^2}}}
Option A (3x−x31−3x2\frac{{3x - {x^3}}}{{1 - 3{x^2}}}) is correct because it is derived from the tangent addition formula and the double angle formula for tangent.
499 Easy
499 of 605
Application +4 | -1
If x+y+z=xyzx + y + z = xyz, then tan⁡−1x+tan⁡−1y+tan⁡−1z={\tan ^{ - 1}}x + {\tan ^{ - 1}}y + {\tan ^{ - 1}}z =
View answer and explanation
Correct option: A
0
Option D (None of these) is correct because the given solution incorrectly concludes that the expression equals 0. The correct simplification leads to a different result.
500 Medium
500 of 605
Application +4 | -1
If 12≤x≤1\frac{1}{2} \leq x \leq 1 then sin⁡−1(3x−4x3)\sin ^{-1}\left(3 x-4 x^3\right) equals -
View answer and explanation
Correct option: B
π−3sin⁡−1x\pi-3 \sin ^{-1} x
Option B (π−3sin⁡−1x\pi - 3 \sin^{-1} x) is correct because when 12≤x≤1\frac{1}{2} \leq x \leq 1, the range of 3sin⁡−1x3 \sin^{-1} x is such that sin⁡−1(3x−4x3)\sin^{-1}(3x - 4x^3) needs to be expressed as π−3sin⁡−1x\pi - 3 \sin^{-1} x to fall within the principal range of sin⁡−1\sin^{-1}.
501 Medium
501 of 605
Application +4 | -1
If the matrix A=[1−11−2]A=\left[\begin{array}{rr}1 & -1 \\ 1 & -2\end{array}\right] satisfies the equation An=5I−8AA^n=5 I-8 A, then nn is equal to
View answer and explanation
Correct option: A
6
Option A (6) is correct because by finding a pattern in the powers of A, we can determine that A6=5I−8AA^6 = 5I - 8A.
502 Easy
502 of 605
Application +4 | -1
The solutions of the equation tan⁡−1(x+1)+tan⁡−1(x−1)=tan⁡−1831{\tan ^{ - 1}}\left( {x + 1} \right) + {\tan ^{ - 1}}\left( {x - 1} \right) = {\tan ^{ - 1}}\frac{8}{{31}} are
View answer and explanation
Correct option: B
14,  −8\frac{1}{4},\; - 8
Option B (14,−8\frac{1}{4}, -8) is correct because these are the solutions obtained after solving the given inverse trigonometric equation using the tangent addition formula.
503 Easy
503 of 605
Application +4 | -1
If 4sin⁡−1x+cos⁡−1x=π,    4{\sin ^{ - 1}}x + {\cos ^{ - 1}}x = \pi ,\;\; then xx is equal to
View answer and explanation
Correct option: B
1/21/2
Option B (1/21/2) is correct because solving the given equation 4sin⁡−1x+cos⁡−1x=π4{\sin ^{ - 1}}x + {\cos ^{ - 1}}x = \pi leads to x=1/2x = 1/2.
504 Medium
504 of 605
Application +4 | -1
If sin⁡−113+sin⁡−123=sin⁡−1x\sin ^{-1} \frac{1}{3}+\sin ^{-1} \frac{2}{3}=\sin ^{-1} x, then xx is equal to:
View answer and explanation
Correct option: C
5+429\frac{\sqrt{5}+4 \sqrt{2}}{9}
Option C (5+429\frac{{\sqrt{{5}}+4 \sqrt{{2}}}}{{9}}) is correct because applying the sine addition formula and simplifying results in this value for x.
505 Medium
505 of 605
Application +4 | -1
The principal value of sin⁡−1(sin⁡10)\sin ^{-1}(\sin 10) is
View answer and explanation
Correct option: C
3π−103 \pi-10
Option C (3π−103\pi - 10) is correct because the principal value of sin⁡−1(sin⁡x)\sin^{-1}(\sin x) must lie in the range [−π2,π2][-\frac{\pi}{2}, \frac{\pi}{2}], and 3π−103\pi - 10 falls within this range.
506 Easy
506 of 605
Application +4 | -1
If cos⁡−1x+cos⁡−1y+cos⁡−1z=π,{\cos ^{ - 1}}x + {\cos ^{ - 1}}y + {\cos ^{ - 1}}z = \pi , then
View answer and explanation
Correct option: C
x2+y2+z2=1−2xyz{x^2} + {y^2} + {z^2} = 1 - 2xyz
Option C (x2+y2+z2=1−2xyz{x^2} + {y^2} + {z^2} = 1 - 2xyz) is correct because it is the simplified form of the given trigonometric equation after applying inverse trigonometric identities and algebraic manipulations.
507 Easy
507 of 605
Application +4 | -1
If the (cos⁡−1x)=sin⁡(cot⁡−112),  ({\cos ^{ - 1}}x) = \sin \left( {{{\cot }^{ - 1}}\frac{1}{2}} \right),\; then xx is equal to
View answer and explanation
Correct option: B
±53\pm \frac{{\sqrt 5 }}{3}
The correct answer is B (±53\pm \frac{{\sqrt 5 }}{3}) because by using trigonometric identities and simplification, we can find the value of x.
508 Medium
508 of 605
Application +4 | -1
The value of cos⁡{tan⁡−1(tan⁡2)}\cos \left\{ {{{\tan }^{ - 1}}(\tan 2)} \right\}, is
View answer and explanation
Correct option: D
−cos⁡2- \cos 2
Option D (−cos⁡2- \cos 2) is correct because by using the property of inverse trigonometric functions, the expression simplifies to −cos⁡2- \cos 2.
509 Easy
509 of 605
Application +4 | -1
If cos⁡−1x+cos⁡−1y=π/2\cos ^{-1} x+\cos ^{-1} y=\pi / 2 and tan⁡−1x−tan⁡−1y=0\tan ^{-1} x-\tan ^{-1} y=0 then x2+xy+y2x^2+x y+y^2 is equal to
View answer and explanation
Correct option: C
3/23 / 2
Option C, 3/23/2, is correct because by using the given conditions, we find that x=y=1/2x = y = 1/\sqrt{2}, and substituting these values into the expression x2+xy+y2x^2 + xy + y^2 gives 3/23/2.
510 Easy
510 of 605
Application +4 | -1
If cos⁡−135−sin⁡−145=cos⁡−1x,{\cos ^{ - 1}}\frac{3}{5} - {\sin ^{ - 1}}\frac{4}{5} = {\cos ^{ - 1}}x, then xx is equal to
View answer and explanation
Correct option: B
1
Option B (1) is correct because simplifying the equation using trigonometric identities leads to cos⁡−1x=0\cos^{-1}x = 0, which implies x = 1.
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