# NCERT solutions for Mathematics Exemplar Class 12 chapter 2 - Inverse Trigonometric Functions [Latest edition]

## Chapter 2: Inverse Trigonometric Functions

Solved ExamplesExercise
Solved Examples [Pages 20 - 35]

### NCERT solutions for Mathematics Exemplar Class 12 Chapter 2 Inverse Trigonometric Functions Solved Examples [Pages 20 - 35]

Solved Examples | Q 1 | Page 20

Find the principal value of cos–1x, for x = sqrt(3)/2.

Solved Examples | Q 2 | Page 21

Evaluate tan^-1(sin((-pi)/2)).

Solved Examples | Q 3 | Page 21

Find the value of cos^-1(cos  (13pi)/6).

Solved Examples | Q 4 | Page 21

Find the value of tan^-1 (tan  (9pi)/8).

Solved Examples | Q 5 | Page 21

Evaluate tan (tan–1(– 4)).

Solved Examples | Q 6 | Page 21

Evaluate: tan^-1 sqrt(3) - sec^-1(-2).

Solved Examples | Q 7 | Page 22

Evaluate: sin^-1 [cos(sin^-1 sqrt(3)/2)]

Solved Examples | Q 8 | Page 22

Prove that tan(cot–1x) = cot(tan–1x). State with reason whether the equality is valid for all values of x.

Solved Examples | Q 9 | Page 22

Find the value of sec(tan^-1  y/2)

Solved Examples | Q 10 | Page 22

Find value of tan (cos–1x) and hence evaluate tan(cos^-1  8/17)

Solved Examples | Q 11 | Page 23

Find the value of sin[2cot^-1 ((-5)/12)]

Solved Examples | Q 12 | Page 23

Evaluate cos[sin^-1  1/4 + sec^-1  4/3]

Solved Examples | Q 13 | Page 24

Prove that 2sin^-1  3/5 - tan^-1  17/31 = pi/4

Solved Examples | Q 14 | Page 24

Prove that cot–17 + cot–18 + cot–118 = cot–13

Solved Examples | Q 15 | Page 25

Which is greater, tan 1 or tan–11?

Solved Examples | Q 16 | Page 25

Find the value of sin(2tan^-1  2/3) + cos(tan^-1 sqrt(3))

Solved Examples | Q 17 | Page 26

Solve for x tan^-1((1 - x)/(1 + x)) = 1/2 tan^-1x, x > 0

Solved Examples | Q 18 | Page 26

Find the values of x which satisfy the equation sin–1x + sin–1(1 – x) = cos–1x.

Solved Examples | Q 19 | Page 26

Solve the equation sin^-1 6x + sin^-1 6sqrt(3)x = - pi/2

Solved Examples | Q 20 | Page 27

Show that 2tan^-1 {tan  alpha/2 * tan(pi/4 - beta/2)} = tan^-1  (sin alpha cos beta)/(cosalpha + sinbeta)

#### Objective type questions Examples 21 to 41

Solved Examples | Q 21 | Page 28

Which of the following corresponds to the principal value branch of tan–1?

• (- pi/2, pi/2)

• [- pi/2, pi/2]

• (- pi/2, pi/2) - {0}

• (0, π)

Solved Examples | Q 22 | Page 28

The principal value branch of sec–1 is ______.

• [- pi/2, pi/2] - {0}

• [0, pi] - {pi/2}

• (0, π)

• (- pi/2, pi/2)

Solved Examples | Q 23 | Page 29

One branch of cos–1 other than the principal value branch corresponds to ______.

• [pi/2, (3pi)/2]

• [pi, 2pi]- {(3pi)/2}

• (0, π)

• [2π, 3π]

Solved Examples | Q 24 | Page 29

The value of sin^-1 (cos((43pi)/5)) is ______.

• (3pi)/5

• (-7pi)/5

• pi/10

• - pi/10

Solved Examples | Q 25 | Page 29

The principal value of the expression cos–1[cos (– 680°)] is ______.

• (2pi)/9

• (-2pi)/9

• (34pi)/9

• pi/9

Solved Examples | Q 26 | Page 29

The value of cot (sin–1x) is ______.

• sqrt(1 + x^2)/x

• x/sqrt(1 + x^2)

• 1/x

• sqrt(1 - x^2)/x

Solved Examples | Q 27 | Page 30

If tan^-1x = pi/10 for some x ∈ R, then the value of cot–1x is ______.

• pi/5

• (2pi)/5

• (3pi)/5

• (4pi)/5

Solved Examples | Q 28 | Page 30

The domain of sin–1 2x is ______.

• [0, 1]

• [– 1, 1]

• [-1/2, 1/2]

• [–2, 2]

Solved Examples | Q 29 | Page 30

The principal value of sin^-1 ((-sqrt(3))/2) is ______.

• - (2pi)/3

• -pi/3

• (4pi)/3

• (5pi)/3

Solved Examples | Q 30 | Page 31

The greatest and least values of (sin–1x)2 + (cos–1x)2 are respectively ______.

• (5pi^2)/4 and pi^2/8

• pi/2 and (-pi)/2

• pi^2/4 ad (-pi^2)/4

• pi^2/4 and 0

Solved Examples | Q 31 | Page 31

Let θ = sin–1 (sin (– 600°), then value of θ is ______.

• pi/3

• pi/2

• (2pi)/3

• (-2pi)/3

Solved Examples | Q 32 | Page 32

The domain of the function y = sin–1 (– x2) is ______.

• [0, 1]

• (0, 1)

• [–1, 1]

• φ

Solved Examples | Q 33 | Page 32

The domain of y = cos–1(x2 – 4) is ______.

• [3, 5]

• [0, π]

• [-sqrt(5), -sqrt(3)] ∩ [-sqrt(5), sqrt(3)]

• [-sqrt(5), -sqrt(3)] ∪ [-sqrt(3), sqrt(5)]

Solved Examples | Q 34 | Page 32

The domain of the function defined by f(x) = sin–1x + cosx is ______.

• [–1, 1]

• [–1, π + 1]

• (– oo, oo)

• φ

Solved Examples | Q 35 | Page 33

The value of sin (2 sin–1 (.6)) is ______.

• .48

• .96

• 1.2

• sin 1.2

Solved Examples | Q 36 | Page 33

If sin–1x + sin–1y = pi/2, then value of cos–1x + cos–1y is ______.

• pi/2

• π

• 0

• (2pi)/3

Solved Examples | Q 37 | Page 33

The value of tan(cos^-1  3/5 + tan^-1  1/4) is ______.

• 19/8

• 8/19

• 19/12

• 3/4

Solved Examples | Q 38 | Page 34

The value of the expression sin [cot–1 (cos (tan–11))] is ______.

• 0

• 1

• 1/sqrt(3)

• sqrt(2/3)

Solved Examples | Q 39 | Page 34

The equation tan–1x – cot–1x = (1/sqrt(3)) has ______.

• No solution

• Unique solution

• Infinite number of solutions

• Two solutions

Solved Examples | Q 40 | Page 34

If α ≤ 2 sin–1x + cos–1x ≤ β, then ______.

• α = (-pi)/2, β = pi/2

• α = β = π

• α = (-pi)/2, β = (3pi)/2

• α = 0, β = 2π

Solved Examples | Q 41 | Page 35

The value of tan2 (sec–12) + cot2 (cosec–13) is ______.

• 5

• 11

• 13

• 15

Exercise [Pages 35 - 41]

### NCERT solutions for Mathematics Exemplar Class 12 Chapter 2 Inverse Trigonometric Functions Exercise [Pages 35 - 41]

Exercise | Q 1 | Page 35

Find the value of tan^-1 (tan  (5pi)/6) +cos^-1(cos  (13pi)/6)

Exercise | Q 2 | Page 35

Evaluate cos[cos^-1 ((-sqrt(3))/2) + pi/6]

Exercise | Q 3 | Page 35

Prove that cot(pi/4 - 2cot^-1 3) = 7

Exercise | Q 4 | Page 35

Find the value of tan^-1 (- 1/sqrt(3)) + cot^-1(1/sqrt(3)) + tan^-1(sin((-pi)/2))

Exercise | Q 5 | Page 35

Find the value of tan^-1 (tan  (2pi)/3)

Exercise | Q 6 | Page 35

Show that 2tan^-1 (-3) = (-pi)/2 + tan^-1 ((-4)/3)

Exercise | Q 7 | Page 36

Find the real solutions of the equation
tan^-1 sqrt(x(x + 1)) + sin^-1 sqrt(x^2 + x + 1) = pi/2

Exercise | Q 8 | Page 36

Find the value of the expression sin(2tan^-1  1/3) + cos(tan^-1 2sqrt(2))

Exercise | Q 9 | Page 36

If 2 tan–1(cos θ) = tan–1(2 cosec θ), then show that θ = π 4, where n is any integer.

Exercise | Q 10 | Page 36

Show that cos(2tan^-1  1/7) = sin(4tan^-1  1/3)

Exercise | Q 11 | Page 36

Solve the following equation cos(tan^-1x) = sin(cot^-1  3/4)

Exercise | Q 12 | Page 36

Prove that tan^-1 ((sqrt(1 + x^2) + sqrt(1 - x^2))/((1 + x^2) - sqrt(1 - x^2))) = pi/2 + 1/2 cos^-1x^2

Exercise | Q 13 | Page 36

Find the simplified form of cos^-1 (3/5 cosx + 4/5 sin x), where x ∈ [(-3pi)/4, pi/4]

Exercise | Q 14 | Page 36

Prove that sin^-1  8/17 + sin^-1  3/5 = sin^-1  7/85

Exercise | Q 15 | Page 36

Show that sin^-1  5/13 + cos^-1  3/5 = tan^-1  63/16

Exercise | Q 16 | Page 36

Prove that tan^-1  1/4 + tan^-1  2/9 = sin^-1  1/sqrt(5)

Exercise | Q 17 | Page 36

Find the value of 4tan^-1  1/5 - tan^-1  1/239

Exercise | Q 18 | Page 37

Show that tan(1/2 sin^-1  3/4) = (4 - sqrt(7))/3 and justify why the other value (4 + sqrt(7))/3 is ignored?

Exercise | Q 19 | Page 37

If a1, a2, a3,...,an is an arithmetic progression with common difference d, then evaluate the following expression.

tan[tan^-1("d"/(1 + "a"_1 "a"_2)) + tan^-1("d"/(21 + "a"_2 "a"_3)) + tan^-1("d"/(1 + "a"_3 "a"_4)) + ... + tan^-1("d"/(1 + "a"_("n" - 1) "a""n"))]

#### Objective Type Questions from 20 to 37

Exercise | Q 20 | Page 37

Which of the following is the principal value branch of cos–1x?

• [(-pi)/2, pi/2]

• (0, π)

• [0, π]

• (0, pi) - {pi/2}

Exercise | Q 21 | Page 37

Which of the following is the principal value branch of cosec–1x?

• ((-pi)/2, pi/2)

• [0, pi] - {pi/2}

• [(-pi)/2, pi/2]

• [(-pi)/2, pi/2] - {0}

Exercise | Q 22 | Page 37

If 3 tan–1x + cot–1x = π, then x equals ______.

• 0

• 1

• – 1

• 1/2

Exercise | Q 23 | Page 37

The value of sin^-1 [cos((33pi)/5)] is ______.

• (3pi)/5

• (-7pi)/5

• pi/10

• (-pi)/10

Exercise | Q 24 | Page 38

The domain of the function cos–1(2x – 1) is ______.

• [0, 1]

• [–1, 1]

• ( –1, 1)

• [0, π]

Exercise | Q 25 | Page 38

The domain of the function defined by f(x) = sin^-1 sqrt(x- 1) is ______.

• [1, 2]

• [–1, 1]

• [0, 1]

• None of these

Exercise | Q 26 | Page 38

If cos(sin^-1  2/5 + cos^-1x) = 0, then x is equal to ______.

• 1/5

• 2/5

• 0

• 1

Exercise | Q 27 | Page 38

The value of sin (2 tan–1(0.75)) is equal to ______.

• 0.75

• 1.5

• 0.96

• sin 1.5

Exercise | Q 28 | Page 38

The value of cos^-1 (cos  (3pi)/2) is equal to ______.

• pi/2

• (3pi)/2

• (5pi)/2

• (7pi)/2

Exercise | Q 29 | Page 38

The value of the expression 2 sec^-1 2 + sin^-1 (1/2) is ______.

• pi/6

• (5pi)/6

• (7pi)/6

• 1

Exercise | Q 30 | Page 38

If tan–1x + tan–1y = (4pi)/5, then cot–1x + cot–1y equals ______.

• pi/5

• (2pi)/5

• (3pi)/5

• π

Exercise | Q 31 | Page 38

If sin^-1 ((2"a")/(1 + "a"^2)) + cos^-1 ((1 - "a"^2)/(1 + "a"^2)) = tan^-1 ((2x)/(1 - x^2)). where a, x ∈ ] 0, 1, then the value of x is ______.

• 0

• "a"/2

• a

• (2"a")/(1 - "a"^2)

Exercise | Q 32 | Page 39

The value of cot[cos^-1 (7/25)] is ______.

• 25/24

• 25/7

• 24/25

• 7/24

Exercise | Q 33 | Page 39

The value of the expression tan (1/2 cos^-1  2/sqrt(5)) is ______.

• 2 + sqrt(5)

• sqrt(5) - 2

• (sqrt(5) + 2)/2

• 5 + sqrt(2)

Exercise | Q 34 | Page 39

If |x| ≤ 1, then 2 tan^-1x + sin^-1 ((2x)/(1 + x^2)) is equal to ______.

• 4 tan^-1x

• 0

• pi/2

• π

Exercise | Q 35 | Page 39

If cos–1α + cos–1β + cos–1γ = 3π, then α(β + γ) + β(γ + α) + γ(α + β) equals ______.

• 0

• 1

• 6

• 12

Exercise | Q 36 | Page 39

The number of real solutions of the equation sqrt(1 + cos 2x) = sqrt(2) cos^-1 (cos x) in [pi/2, pi] is ______.

• 0

• 1

• 2

• Infinite

Exercise | Q 37 | Page 39

If cos–1x > sin–1x, then ______.

• 1/sqrt(2) < x ≤ 1

• 0 ≤ x < 1/2

• -1 ≤ x  < 1/2

• x > 0

#### Fill in the blanks 38 to 48

Exercise | Q 38 | Page 40

The principal value of cos^-1 (- 1/2) is ______.

Exercise | Q 39 | Page 40

The value of sin^-1 (sin  (3pi)/5) is ______.

Exercise | Q 40 | Page 40

If cos(tan^-1x + cot^-1 sqrt(3)) = 0, then value of x is ______.

Exercise | Q 41 | Page 40

The set of values of sec^-1 (1/2) is ______.

Exercise | Q 42 | Page 40

The principal value of tan^-1 sqrt(3) is ______.

Exercise | Q 43 | Page 40

The value of cos^-1 (cos  (14pi)/3) is ______.

Exercise | Q 44 | Page 40

The value of cos (sin–1x + cos–1x), |x| ≤ 1 is ______.

Exercise | Q 45 | Page 40

The value of expression tan((sin^-1x + cos^-1x)/2), when x = sqrt(3)/2 is ______.

Exercise | Q 46 | Page 40

If y = 2 tan^-1x + sin^-1 ((2x)/(1 + x^2)) for all x, then ______ < y < ______.

Exercise | Q 47 | Page 40

The result tan^1x - tan^-1y = tan^-1 ((x - y)/(1 + xy)) is true when value of xy is ______.

Exercise | Q 48 | Page 40

The value of cot–1(–x) for all x ∈ R in terms of cot–1x is ______.

#### State True or False for the statement 49 to 55

Exercise | Q 49 | Page 40

All trigonometric functions have inverse over their respective domains.

• True

• False

Exercise | Q 50 | Page 40

The value of the expression (cos–1x)2 is equal to sec2x.

• True

• False

Exercise | Q 51 | Page 40

The domain of trigonometric functions can be restricted to any one of their branch (not necessarily principal value) in order to obtain their inverse functions.

• True

• False

Exercise | Q 52 | Page 40

The least numerical value, either positive or negative of angle θ is called principal value of the inverse trigonometric function.

• True

• False

Exercise | Q 53 | Page 40

The graph of inverse trigonometric function can be obtained from the graph of their corresponding trigonometric function by interchanging x and y axes.

• True

• False

Exercise | Q 54 | Page 41

The minimum value of n for which tan^-1  "n"/pi > pi/4, n ∈ N, is valid is 5.

• True

• False

Exercise | Q 55 | Page 41

The principal value of sin^-1 [cos(sin^-1  1/2)] is pi/3.

• True

• False

## Chapter 2: Inverse Trigonometric Functions

Solved ExamplesExercise

## NCERT solutions for Mathematics Exemplar Class 12 chapter 2 - Inverse Trigonometric Functions

NCERT solutions for Mathematics Exemplar Class 12 chapter 2 (Inverse Trigonometric Functions) include all questions with solution and detail explanation. This will clear students doubts about any question and improve application skills while preparing for board exams. The detailed, step-by-step solutions will help you understand the concepts better and clear your confusions, if any. Shaalaa.com has the CBSE Mathematics Exemplar Class 12 solutions in a manner that help students grasp basic concepts better and faster.

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Concepts covered in Mathematics Exemplar Class 12 chapter 2 Inverse Trigonometric Functions are Inverse Trigonometric Functions (Simplification and Examples), Properties of Inverse Trigonometric Functions, Graphs of Inverse Trigonometric Functions, Inverse Trigonometric Functions - Principal Value Branch, Basic Concepts of Trigonometric Functions.

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