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JEE • MATHEMATICS • INVERSE TRIGONOMETRY

Inverse trigonometry practice · Mixed difficulty

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571 Easy
571 of 605
Application +4 | -1
If tan⁡−1(ax)+tan⁡−1(bx)=π2{\tan ^{ - 1}}\left( {\frac{a}{x}} \right) + {\tan ^{ - 1}}\left( {\frac{b}{x}} \right) = \frac{\pi }{2}, then xx is equal to
View answer and explanation
Correct option: A
ab\sqrt {ab}
Option A (ab\sqrt {ab}) is correct because it is the solution obtained by applying the tangent addition formula and solving the resulting equation.
572 Easy
572 of 605
Application +4 | -1
The value of cos⁡[2tan⁡−1(−7)]\cos \left[ {2{{\tan }^{ - 1}}\left( { - 7} \right)} \right] is
View answer and explanation
Correct option: D
−2425- \frac{{24}}{{25}}
Option D (−2425- \frac{{24}}{{25}}) is correct because it is the result of applying the double angle formula for cosine in terms of tangent and simplifying the expression.
573 Medium
573 of 605
Application +4 | -1
The value of cos⁡−1(cot⁡(sin⁡−1(2−34)+cos⁡−1(124)+sec⁡−1(2)))\cos ^{-1}\left(\cot \left(\sin ^{-1}\left(\sqrt{\frac{2-\sqrt{3}}{4}}\right)+\cos ^{-1}\left(\frac{\sqrt{12}}{4}\right)+\sec ^{-1}(\sqrt{2})\right)\right) is
View answer and explanation
Correct option: D
π2\frac{\pi}{2}
Option D (π2\frac{\pi}{2}) is correct because the expression simplifies to cos⁡−1(0)\cos^{-1}(0), which equals π2\frac{\pi}{2}.
574 Easy
574 of 605
Analyzing +4 | -1
If cos⁡−1x+cos⁡−1y=π2  {\cos ^{ - 1}}x + {\cos ^{ - 1}}y = \frac{\pi }{2}\; and tan⁡−1x−tan⁡−1y=0,  {\tan ^{ - 1}}x - {\tan ^{ - 1}}y = 0,\; then x2+xy+y2  {x^2} + xy + {y^2}\; is equal to
View answer and explanation
Correct option: C
32\frac{3}{2}
Option C (3/2) is correct because from the given equations, we can deduce that x = y = 1/sqrt(2), and substituting these values into the expression x² + xy + y² yields 3/2.
575 Medium
575 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: B
x2+y2+z2+2xyz=1x^2+y^2+z^2+2 x y z=1
Option B (x2+y2+z2+2xyz=1x^2+y^2+z^2+2xyz=1) is correct because using the identity cos⁡−1x+cos⁡−1y+cos⁡−1z=π\cos^{-1} x + \cos^{-1} y + \cos^{-1} z = \pi and simplifying the expression leads to the correct equation.
576 Easy
576 of 605
Application +4 | -1
If sin⁡(sin⁡−115+cos⁡−1x)=1\sin \left( {{{\sin }^{ - 1}}\frac{1}{5} + {{\cos }^{ - 1}}x} \right) = 1, then xx is equal to
View answer and explanation
Correct option: D
1/51/5
Option D is correct because solving the equation sin⁡(sin⁡−115+cos⁡−1x)=1\sin \left( {{{\sin }^{ - 1}}\frac{1}{5} + {{\cos }^{ - 1}}x} \right) = 1 using trigonometric identities leads to x=15x = \frac{1}{5}.
577 Easy
577 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}
Option B (±53\pm \frac{{\sqrt 5 }}{3}) is correct because by setting cot⁡−112=ϕ{\cot ^{ - 1}}\frac{1}{2} = \phi and cos⁡−1x=θ{\cos ^{ - 1}}x = \mathrm{\theta }, we can solve for x using trigonometric identities.
578 Medium
578 of 605
Analyzing +4 | -1
The number of solutions of the equation cos⁡−1(1−x)+mcos⁡−1(x)=nπ2\cos ^{-1}(1-x)+m \cos ^{-1}(x)=\frac{n \pi}{2}, where m>0m > 0, n≤0n \leq 0, is
View answer and explanation
Correct option: A
0
Option A (0) is correct because the limit from both sides (x → 0+ and x → 0-) results in [sin(1)/1] = 0 and [sin(-1)/-1] = 0, respectively.
579 Easy
579 of 605
Application +4 | -1
cosec[2cot⁡−1(5)+cos⁡−1(45)]\mathrm{cosec}\left[ {2{{\cot }^{ - 1}}(5) + {{\cos }^{ - 1}}\left( {\frac{4}{5}} \right)} \right] is equal to
View answer and explanation
Correct option: C
6556 \frac{{65}}{{56}}\,
The correct answer is C (6556\frac{{65}}{{56}}) because it correctly simplifies the trigonometric expression using inverse trigonometric identities.
580 Medium
580 of 605
Application +4 | -1
tan⁡−1(xy)−tan⁡−1(x−yx+y)\tan ^{-1}\left(\frac{x}{y}\right)-\tan ^{-1}\left(\frac{x-y}{x+y}\right) is
View answer and explanation
Correct option: D
π4\frac{\pi}{4} or −3π4-\frac{3 \pi}{4}
Option D (π4\frac{{\pi}}{{4}} or −3π4-\frac{{3 \pi}}{{4}}) is correct because simplifying the expression using tangent subtraction and properties of inverse trigonometric functions yields these two possible values.
581 Easy
581 of 605
Analyzing +4 | -1
If ∑i=120sin⁡−1xi=10π,\sum \limits_{i = 1}^{20} {\sin ^{ - 1}}{x_i} = 10\pi , then ∑i=120xi\sum \limits_{i = 1}^{20} {x_i} is equal to
View answer and explanation
Correct option: A
20
Option A (20) is correct because the maximum value of sin⁡−1x{\sin ^{ - 1}}x is π2\frac{\pi}{2}. Therefore, each term in the summation must be π2\frac{\pi}{2}, implying each xix_i is 1.
582 Easy
582 of 605
Application +4 | -1
If tan⁡−1(x−1)+tan⁡−1x+tan⁡−1(x+1)=tan⁡−13x{\tan ^{ - 1}}\left( {x - 1} \right) + {\tan ^{ - 1}}x + {\tan ^{ - 1}}\left( {x + 1} \right) = {\tan ^{ - 1}}3x, then xx is
View answer and explanation
Correct option: D
0,  ±120,\; \pm \frac{1}{2}
Option D (0,±120, \pm \frac{1}{2}) is correct because these are the solutions obtained by simplifying the given equation using the arctangent addition formula.
583 Medium
583 of 605
Application +4 | -1
cos⁡[cos⁡−1(−17)+sin⁡−1(−17)]\cos \left[ {{{\cos }^{ - 1}}\left( { - \frac{1}{7}} \right) + {{\sin }^{ - 1}}\left( { - \frac{1}{7}} \right)} \right] is equal to
View answer and explanation
Correct option: B
0
Option B (0) is correct because the expression simplifies to cos(π/2), which equals 0.
584 Hard
584 of 605
Analyzing +4 | -1
Domain of the function f(x)=2Sin⁡−11−x+Sin⁡−1[2x(1−x)]f(x)=2 \operatorname{Sin}^{-1} \sqrt{1-x}+\operatorname{Sin}^{-1}[2 \sqrt{x(1-x)}] is
View answer and explanation
Correct option: A
0≤x≤10 \leq x \leq 1
Option A (0≤x≤10 \leq x \leq 1) is correct because it satisfies the domain restrictions of both inverse sine functions, ensuring that the arguments are within [-1, 1].
585 Easy
585 of 605
Application +4 | -1
If tan⁡−1x−tan⁡−1y=tan⁡−1A,then  A  is  equal  to{\tan ^{ - 1}}x - {\tan ^{ - 1}}y = {\tan ^{ - 1}}A,\mathrm{then\;}A\mathrm{\;is\;equal\;to}
View answer and explanation
Correct option: C
x−y1+xy\frac{{x - y}}{{1 + xy}}
Option C (x−y1+xy\frac{{x - y}}{{1 + xy}}) is correct because it directly applies the tangent subtraction formula: tan⁡−1x−tan⁡−1y=tan⁡−1(x−y1+xy){\tan ^{ - 1}}x - {\tan ^{ - 1}}y = {\tan ^{ - 1}}\left( {\frac{{x - y}}{{1 + xy}}} \right).

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586 Easy
586 of 605
Application +4 | -1
If sin⁡(sin⁡−115+cos⁡−1x)=1\sin \left( {{{\sin }^{ - 1}}\frac{1}{5} + {{\cos }^{ - 1}}x} \right) = 1, then the value of xx is
View answer and explanation
Correct option: D
1/51/5
Option D (1/5) is correct because it is derived from using the identity sin⁡−1x+cos⁡−1x=π2\sin^{-1}x + \cos^{-1}x = \frac{\pi}{2} and solving for x.
587 Easy
587 of 605
Analyzing +4 | -1
If sin⁡−1(2x1+x2)+cos⁡−1(1−x21+x2)=4tan⁡−1x{\sin ^{ - 1}}\left( {\frac{{2x}}{{1 + {x^2}}}} \right) + {\cos ^{ - 1}}\left( {\frac{{1 - {x^2}}}{{1 + {x^2}}}} \right) = 4{\tan ^{ - 1}}x, then
View answer and explanation
Correct option: C
x∈[0,  1]x \in \left[ {0,\;1} \right]
Option C (x∈[0,  1]x \in \left[ {0,\;1} \right]) is correct because it is the interval where both inverse trigonometric functions are defined and the equation holds true.
588 Medium
588 of 605
Application +4 | -1
A root of the equation 17x2+17xtan⁡[2tan⁡−1(1/5)−π/4]−10=017 x^2+17 x \tan \left[2 \tan ^{-1}(1 / 5)-\pi / 4\right]-10=0 is
View answer and explanation
Correct option: D
1
Option D (1) is correct because by simplifying the expression inside the tangent function and substituting it back into the equation, we find that x=1x=1 is a root.
589 Easy
589 of 605
Analyzing +4 | -1
Which one of the following is correct?
View answer and explanation
Correct option: A
tan⁡1>tan⁡−11\tan 1 > {\tan ^{ - 1}}1
Option A (tan⁡1>tan⁡−11\tan 1 > {\tan ^{ - 1}}1) is correct because 1 radian is greater than 45 degrees, so tan⁡1>1\tan 1 > 1, and tan⁡−11=π4<1{\tan ^{ - 1}}1 = \frac{\pi }{4} < 1.
590 Easy
590 of 605
Application +4 | -1
A value of xx for which sin⁡(cot⁡−1(1+x))=cos⁡(tan⁡−1x)\sin \left(\cot ^{-1}(1+x)\right)=\cos \left(\tan ^{-1} x\right), is
View answer and explanation
Correct option: A
−12-\frac{1}{2}
Option A (−12-\frac{{1}}{{2}}) is correct because substituting x=−12x = -\frac{{1}}{{2}} into the given equation satisfies the equation after simplification.
591 Easy
591 of 605
Analyzing +4 | -1
If the mapping f(x)=ax+b,a>0f\left( x \right) = ax + b,a > 0 maps [−1,1- 1,1] onto [0, 2] then cot⁡[cot⁡−17+cot⁡−18+cot⁡−118]\cot [{\cot ^{ - 1}}7 + {\cot ^{ - 1}}8 + {\cot ^{ - 1}}18] is equal to
View answer and explanation
Correct option: D
f(2)f\left( 2 \right)
Option D (f(2)f\left( 2 \right)) is correct because after finding the function f(x) and simplifying the cotangent expression, the result equals 3, which is f(2).
592 Easy
592 of 605
Application +4 | -1
tan⁡−13−cot⁡−1(−3)\tan ^{-1} \sqrt{3}-\cot ^{-1}(-\sqrt{3}) is equal to
View answer and explanation
Correct option: B
−π2-\frac{\pi}{2}
Option B (−π2-\frac{{\pi}}{{2}}) is correct because it is the result of subtracting the principal values of the inverse cotangent from the inverse tangent.
593 Easy
593 of 605
Application +4 | -1
sin⁡[3sin⁡−1(15)]\sin \left[ {3{{\sin }^{ - 1}}\left( {\frac{1}{5}} \right)} \right] is equal to
View answer and explanation
Correct option: A
71125\frac{{71}}{{125}}
Option A (71/125) is correct because it is the result of applying the triple angle identity for sine and simplifying the expression.
594 Medium
594 of 605
Application +4 | -1
Let A={1,2,3,4}A=\{1,2,3,4\} and x,y∈Ax, y \in A, then the value of max⁡(sin⁡−1(sin⁡x)+cos⁡−1(cos⁡y))\max \left(\sin ^{-1}(\sin x)+\cos ^{-1}(\cos y)\right) is
View answer and explanation
Correct option: B
π+1\pi+1
Option D ((4,25)(4,2 \sqrt{5})) is correct because the point lies on the director circle of the given circle, ensuring the tangents drawn from that point are at right angles.
595 Medium
595 of 605
Application +4 | -1
sin⁡−1sin⁡(16)\sin ^{-1} \sin (16) is equal to
View answer and explanation
Correct option: A
5π−165 \pi-16
Option A (5π−165 \pi-16) is the correct answer because it falls within the principal range of the inverse sine function after applying trigonometric identities.
596 Medium
596 of 605
Application +4 | -1
If (tan⁡−1x)2+(cos⁡−1x)2=5π28\left(\tan ^{-1} x\right)^2+\left(\cos ^{-1} x\right)^2=\frac{5 \pi^2}{8}, then xx equals
View answer and explanation
Correct option: A
−1-1
Option A (-1) is correct because it satisfies the given equation after using the identity relating tan⁡−1x\tan^{-1} x and cos⁡−1x\cos^{-1} x.
597 Easy
597 of 605
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+z100−9x101+y101+z101{x^{100}} + {y^{100}} + {z^{100}} - \frac{9}{{{x^{101}} + {y^{101}} + {z^{101}}}} is
View answer and explanation
Correct option: A
0
Option A (0) is correct because if sin⁡−1x+sin⁡−1y+sin⁡−1z=3π2{\sin ^{ - 1}}x + {\sin ^{ - 1}}y + {\sin ^{ - 1}}z = \frac{{3\pi }}{2}, then x=y=z=1x=y=z=1, and substituting these values into the expression yields 0.
598 Easy
598 of 605
Application +4 | -1
If the inverse trigonometric functions take principal values, then cos⁡−1(310cos⁡(tan⁡−1(43))+25sin⁡(tan⁡−1(43))){\cos ^{ - 1}}\left( {\frac{3}{{10}}\cos \left( {{{\tan }^{ - 1}}\left( {\frac{4}{3}} \right)} \right) + \frac{2}{5}\sin \left( {{{\tan }^{ - 1}}\left( {\frac{4}{3}} \right)} \right)} \right) is equal to:
View answer and explanation
Correct option: C
π3\frac{\pi }{3}
Option C (π3\frac{\pi }{3}) is correct because after simplifying the trigonometric expressions inside the inverse cosine, the result is cos⁻¹(1/2), which equals π/3.
599 Medium
599 of 605
Application +4 | -1
The number of positive integral solutions of the equation tan⁡−1x+cos⁡−1y1+y2=sin⁡−1310{\tan ^{ - 1}}x + {\cos ^{ - 1}}\frac{y}{{\sqrt {1 + {y^2}} }} = {\sin ^{ - 1}}\frac{3}{{\sqrt {10} }} is
View answer and explanation
Correct option: B
Two
Option B (Two) is correct because there are two positive integral solutions (1,2) and (2,7) that satisfy the given equation after simplification and trial.
600 Easy
600 of 605
Application +4 | -1
The value of sin⁡−1{cos⁡(4095∘)}{\sin ^{ - 1}}\left\{ {\cos \left( {4095^\circ } \right)} \right\} is equal to
View answer and explanation
Correct option: C
−π4- \frac{\pi }{4}
Option C (−π4- \frac{\pi }{4}) is correct because it is the simplified value of the given expression after applying trigonometric identities and considering the range of the inverse sine function.
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