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JEE • MATHEMATICS • INVERSE TRIGONOMETRY

Inverse trigonometry practice · Mixed difficulty

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331 Medium
331 of 605
Application +4 | -1
The value of sin⁡(cot⁡−1(cos⁡(tan⁡−1x))\sin \left(\cot ^{-1}\left(\cos \left(\tan ^{-1} x\right)\right)\right. is
View answer and explanation
Correct option: B
x2+1x2+2\sqrt{\frac{x^2+1}{x^2+2}}
Option B (x2+1x2+2\sqrt{\frac{x^2+1}{x^2+2}}) is correct because the expression simplifies using trigonometric identities and right triangle relationships.
332 Medium
332 of 605
Analyzing +4 | -1
The xx satisfying sin⁡−1x+sin⁡−1(1−x)=cos⁡−1x\sin ^{-1} x+\sin ^{-1}(1-x)=\cos ^{-1} x are
View answer and explanation
Correct option: C
0,120, \frac{1}{2}
Option C (0, 1/2) is correct because these are the solutions to the equation obtained after applying trigonometric identities and solving the resulting algebraic equation.
333 Easy
333 of 605
Application +4 | -1
5cos⁡−1(1−x21+x2)+7sin⁡−1(2x1+x2)−4tan⁡−1(2x1−x2)−tan⁡−1x=5π5{\cos ^{ - 1}}\left( {\frac{{1 - {x^2}}}{{1 + {x^2}}}} \right) + 7{\sin ^{ - 1}}\left( {\frac{{2x}}{{1 + {x^2}}}} \right) - 4{\tan ^{ - 1}}\left( {\frac{{2x}}{{1 - {x^2}}}} \right) - {\tan ^{ - 1}}x = 5\pi, then xx is equal to
View answer and explanation
Correct option: D
3\sqrt 3
Option D (3\sqrt 3) is the correct answer because substituting x=3x = \sqrt{3} into the given equation satisfies the equation.
334 Medium
334 of 605
Application +4 | -1
The equation sin⁡−1x=2sin⁡−1a\sin ^{-1} x=2 \sin ^{-1} a holds true for
View answer and explanation
Correct option: C
−12≤a≤12-\frac{1}{\sqrt{2}} \leq a \leq \frac{1}{\sqrt{2}}
Option C (−12≤a≤12-\frac{{1}}{{\sqrt{{2}}}} \leq a \leq \frac{{1}}{{\sqrt{{2}}}}) is correct because it defines the range of 'a' for which the equation sin⁡−1x=2sin⁡−1a\sin^{-1} x = 2 \sin^{-1} a holds true, considering the domain and range of inverse trigonometric functions.
335 Easy
335 of 605
Application +4 | -1
The value of cos⁡−1(cos⁡5π3)+sin⁡−1(cos⁡5π3){\cos ^{ - 1}}\left( {\cos \frac{{5\pi }}{3}} \right) + {\sin ^{ - 1}}\left( {\cos \frac{{5\pi }}{3}} \right) is
View answer and explanation
Correct option: C
π2\frac{\pi }{2}
Option C (π2\frac{\pi}{2}) is correct because cos⁡−1x+sin⁡−1x=π2{\cos ^{ - 1}}x + {\sin ^{ - 1}}x = \frac{\pi}{2} for any xx in the domain.
336 Medium
336 of 605
Application +4 | -1
tan⁡−1(tan⁡4)+cot⁡−1(cot⁡4)\tan ^{-1}(\tan 4)+\cot ^{-1}(\cot 4) equals
View answer and explanation
Correct option: C
8−2π8-2 \pi
Option C (8−2π8 - 2\pi) is correct because, due to the range restrictions of tan⁡−1\tan^{-1} and cot⁡−1\cot^{-1}, the expression simplifies to 8−2π8 - 2\pi.
337 Medium
337 of 605
Application +4 | -1
If cos⁡−1x+cos⁡−1y+cos⁡−1z=3π\cos ^{-1} x+\cos ^{-1} y+\cos ^{-1} z=3 \pi then xy+yz+zxx y+y z+z x is equal to :
View answer and explanation
Correct option: D
3
Option D is correct because when cos⁡−1x+cos⁡−1y+cos⁡−1z=3π\cos^{-1}x + \cos^{-1}y + \cos^{-1}z = 3\pi, it implies x=y=z=−1x=y=z=-1, and therefore xy+yz+zx=3xy+yz+zx = 3.
338 Easy
338 of 605
Application +4 | -1
Solution of the equation cot⁡−1x+sin⁡−115=π4{\cot ^{ - 1}}x + {\sin ^{ - 1}}\frac{1}{{\sqrt 5 }} = \frac{\pi }{4} is
View answer and explanation
Correct option: A
x=3x = 3
Option A (x = 3) is correct because solving the equation using the arctangent addition formula leads to x = 3.
339 Medium
339 of 605
Application +4 | -1
Let x∈(0,1)x \in(0,1). The set of all xx such that sin⁡−1x>cos⁡−1x\sin ^{-1} x>\cos ^{-1} x, is the interval:
View answer and explanation
Correct option: B
(12,1)\left(\frac{1}{\sqrt{2}}, 1\right)
Option B ((12,1)\left(\frac{{1}}{{\sqrt{{2}}}}, 1\right)) is correct because solving the inequality sin⁡−1x>cos⁡−1x\sin ^{-1} x > \cos ^{-1} x leads to x>12x > \frac{{1}}{{\sqrt{{2}}}} and x<1x < 1.
340 Medium
340 of 605
Analyzing +4 | -1
The number of real solutions of the equation tan⁡−1x2−3x+2+cos⁡−14x−x2−3=π\tan ^{-1} \sqrt{x^2-3 x+2}+\cos ^{-1} \sqrt{4 x-x^2-3}=\pi is
View answer and explanation
Correct option: C
Zero
Option C (Zero) is correct because the given equation has no real solutions as the range of the left-hand side is strictly less than π\pi.
341 Easy
341 of 605
Application +4 | -1
Value of tan⁡−1(sin⁡2−1cos⁡2){\tan ^{ - 1}}\left( {\frac{{\sin 2 - 1}}{{\cos 2}}} \right) is
View answer and explanation
Correct option: B
1−π41 - \frac{\pi }{4}
Option B (1−π41 - \frac{\pi }{4}) is correct because by applying trigonometric identities and simplifying the expression, we find that tan⁡−1(sin⁡2−1cos⁡2)=1−π4{\tan ^{ - 1}}\left( {\frac{{\sin 2 - 1}}{{\cos 2}}} \right) = 1 - \frac{\pi }{4}.
342 Easy
342 of 605
Application +4 | -1
The value of sin⁡[π2−sin⁡−1  (−32)]\sin \left[ {\frac{\pi }{2} - {{\sin }^{ - 1}}\;\left( { - \frac{{\sqrt 3 }}{2}} \right)} \right] is
View answer and explanation
Correct option: C
12\frac{1}{2}
Option C (12\frac{1}{2}) is correct because using the complementary angle identity and simplifying gives sin⁡[π2−sin⁡−1(−32)]=12\sin \left[ {\frac{\pi}{2} - {{\sin }^{ - 1}}\left( { - \frac{{\sqrt 3 }}{2}} \right)} \right] = \frac{1}{2}.
343 Medium
343 of 605
Application +4 | -1
tan⁡[2tan⁡−1(15)−π4]\tan \left[2 \tan ^{-1}\left(\frac{1}{5}\right)-\frac{\pi}{4}\right] is equal to
View answer and explanation
Correct option: D
−717-\frac{7}{17}
Option D (−717-\frac{7}{17}) is correct because by applying the double angle formula for tangent and then the tangent subtraction formula, we arrive at the correct value.
344 Easy
344 of 605
Application +4 | -1
tan⁡[12sin⁡−1(2a1+a2)+12cos⁡−1(1−a21+a2)]  \tan \left[ {\frac{1}{2}{{\sin }^{ - 1}}\left( {\frac{{2a}}{{1 + {a^2}}}} \right) + \frac{1}{2}{{\cos }^{ - 1}}\left( {\frac{{1 - {a^2}}}{{1 + {a^2}}}} \right)} \right]\; is equal to
View answer and explanation
Correct option: C
2a1−a2\frac{{2a}}{{1 - {a^2}}}
Option C (2a1−a2\frac{{2a}}{{1 - {a^2}}}) is correct because the expression simplifies to tan⁡(2tan⁡−1a)\tan(2{\tan ^{ - 1}}a), which equals 2a1−a2\frac{{2a}}{{1 - {a^2}}}.
345 Medium
345 of 605
Application +4 | -1
Assertion: 2tan⁡−112+tan⁡−117=tan⁡−131172 \tan ^{-1} \frac{1}{2}+\tan ^{-1} \frac{1}{7}=\tan ^{-1} \frac{31}{17}.

Reason: 2tan⁡−1x=tan⁡−1(2x1−x2)2 \tan ^{-1} x=\tan ^{-1}\left(\frac{2 x}{1-x^2}\right) if −1<x<1-1 < x < 1.
View answer and explanation
Correct option: A
Assertion is correct, reason is correct; reason is a correct explanation for assertion.
Option A is correct because the assertion is true, the reason is true, and the reason correctly explains the assertion using the trigonometric identity for 2tan⁡−1x2 \tan^{-1}x.

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346 Easy
346 of 605
Application +4 | -1
sin⁡−1(4/5)+sin⁡−1(5/13)+sin⁡−1(16/65)\sin ^{-1}(4 / 5)+\sin ^{-1}(5 / 13)+\sin ^{-1}(16 / 65) is equal to
View answer and explanation
Correct option: B
π/2\pi / 2
Option B (π/2\pi/2) is correct because the sum of the inverse sines simplifies to sin⁡−1(63/65)+sin⁡−1(16/65)\sin^{-1}(63/65) + \sin^{-1}(16/65), which equals π/2\pi/2 due to the identity sin⁡−1x+cos⁡−1x=π/2\sin^{-1}x + \cos^{-1}x = \pi/2.
347 Easy
347 of 605
Application +4 | -1
The value of tan⁡−1(1)+cos⁡−1(−12)+sin⁡−1(−12)\tan ^{-1}(1)+\cos ^{-1}\left(-\frac{1}{2}\right)+\sin ^{-1}\left(-\frac{1}{2}\right) is equal to
View answer and explanation
Correct option: C
3π4\frac{3 \pi}{4}
Option C (3π\pi/4) is correct because it is the result of evaluating the inverse trigonometric functions and summing the results.
348 Medium
348 of 605
Application +4 | -1
If θ=cot⁡−17+cot⁡−18+cot⁡−118\theta = {\cot ^{ - 1}}7 + {\cot ^{ - 1}}8 + {\cot ^{ - 1}}18, then the value of cot⁡ θ\cot \,\theta must be
View answer and explanation
Correct option: A
3
Option A (3) is correct because by converting the cotangent inverse to tangent inverse and using the tangent addition formula, we find that cot⁡θ=3\cot \theta = 3.
349 Easy
349 of 605
Application +4 | -1
If sin⁡−1(1−x)−2sin⁡−1x=π2{\sin ^{ - 1}}\left( {1 - x} \right) - 2{\sin ^{ - 1}}x = \frac{\pi }{2}, then xx equals
View answer and explanation
Correct option: C
0
Option C (0) is correct because, after solving the equation, x=0x = 0 is the only solution that satisfies the original equation.
350 Easy
350 of 605
Analyzing +4 | -1
If α=sin⁡−132+sin⁡−113,  β=cos⁡−132+cos⁡−113\alpha = {\sin ^{ - 1}}\frac{{\sqrt 3 }}{2} + {\sin ^{ - 1}}\frac{1}{3},\;\beta = {\cos ^{ - 1}}\frac{{\sqrt 3 }}{2} + {\cos ^{ - 1}}\frac{1}{3}, then
View answer and explanation
Correct option: C
α<β\alpha < \beta
Option C (α<β\alpha < \beta) is correct because, given α+β=π\alpha + \beta = \pi, and by comparing the values of α\alpha and π2\frac{\pi}{2}, it can be shown that β>π2\beta > \frac{\pi}{2}, thus α<β\alpha < \beta.
351 Easy
351 of 605
Application +4 | -1
The value of cot⁡(cosec−153+tan⁡−123)\cot \left( {\mathrm{cose}{\mathrm{c}^{ - 1}}\frac{5}{3} + {{\tan }^{ - 1}}\frac{2}{3}} \right) is
View answer and explanation
Correct option: B
617\frac{6}{{17}}
Option B (617\frac{6}{{17}}) is correct because it is the simplified value of the given expression after applying trigonometric identities and the cotangent addition formula.
352 Medium
352 of 605
Application +4 | -1
If a=sin⁡−1(sin⁡(5))a=\sin ^{-1}(\sin (5)) and b=cos⁡−1(cos⁡(5))b=\cos ^{-1}(\cos (5)), then a2+b2a^2+b^2 is equal to
View answer and explanation
Correct option: B
8π2−40π+508 \pi^2-40 \pi+50
Option A (8π2−40π+508 \pi^2-40 \pi+50) is correct because it is the correct simplification after finding a and b using the given conditions.
353 Easy
353 of 605
Application +4 | -1
The value of cos⁡−1(cos⁡5π3)+sin⁡−1(sin⁡5π3)\cos ^{-1}\left(\cos \frac{5 \pi}{3}\right)+\sin ^{-1}\left(\sin \frac{5 \pi}{3}\right) is
View answer and explanation
Correct option: A
0
Option A (0) is correct because after simplifying using trigonometric properties and the ranges of inverse trigonometric functions, the expression equals 0.
354 Easy
354 of 605
Application +4 | -1
If tan⁡−11+x2−1x=4∘\tan ^{-1} \frac{\sqrt{1+x^2}-1}{x}=4^{\circ}, then
View answer and explanation
Correct option: D
x=tan⁡8∘x=\tan 8^{\circ}
Option D (x=tan⁡8∘x=\tan 8^{\circ}) is correct because substituting x=tan⁡θx = \tan \theta and simplifying using trigonometric identities leads to θ=8∘\theta = 8^{\circ}.
355 Medium
355 of 605
Analyzing +4 | -1
The equation 2cos⁡−1x+sin⁡−1x=11π62{\cos ^{ - 1}}x + {\sin ^{ - 1}}x = \frac{{11\pi }}{6} has
View answer and explanation
Correct option: A
No solution
Option A (No solution) is correct because the equation leads to a value of cos⁡−1x{\cos ^{ - 1}}x that is outside its defined range of [0, π].
356 Easy
356 of 605
Application +4 | -1
2tan⁡−1(13)+tan⁡−1(17)2{\tan ^{ - 1}}\left( {\frac{1}{3}} \right) + {\tan ^{ - 1}}\left( {\frac{1}{7}} \right) is equal to
View answer and explanation
Correct option: D
π4\frac{\pi }{4}
Option D (π4\frac{\pi}{4}) is correct because using the arctangent addition formula twice, the expression simplifies to π4\frac{\pi}{4}.
357 Easy
357 of 605
Application +4 | -1
The principle value of sin⁡−1tan⁡(−5π4){\sin ^{ - 1}}\tan \left( {\frac{{ - 5\pi }}{4}} \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 sin⁡−1(−1){\sin ^{ - 1}}(-1), which equals −π2- \frac{\pi }{{2}}.
358 Easy
358 of 605
Application +4 | -1
If sin⁡−1(12)+cosec⁡−1(2)\sin ^{-1}\left(\frac{1}{\sqrt{2}}\right)+\operatorname{cosec}^{-1}(2) has the value kπ12{k} \frac{\pi}{12}, then value of kk is
View answer and explanation
Correct option: A
5
Option A (5) is correct because by evaluating the inverse trigonometric functions and simplifying, the value of k is found to be 5.
359 Medium
359 of 605
Application +4 | -1
The value of cos⁡[tan⁡−1(tan⁡15π4)]\cos \left[\tan ^{-1}\left(\tan \frac{15 \pi}{4}\right)\right] is
View answer and explanation
Correct option: A
12\frac{1}{\sqrt{2}}
Option A (12\frac{{1}}{{\sqrt{{2}}}}) is correct because it is obtained by simplifying the expression using the properties of tangent and inverse trigonometric functions.
360 Medium
360 of 605
Application +4 | -1
The value of cot⁡−134+sin⁡−1513{\cot ^{ - 1}}\frac{3}{4} + {\sin ^{ - 1}}\frac{5}{{13}} is
View answer and explanation
Correct option: A
sin⁡−16365{\sin ^{ - 1}}\frac{{63}}{{65}}
Option A is correct because by converting both terms to arcsin and using the arcsin addition formula, the expression simplifies to sin⁡−16365{\sin ^{ - 1}}\frac{{63}}{{65}}.
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