RD Sharma Solutions For Class 12 Maths Exercise 4.14 Chapter 4 Inverse Trigonometric Functions

Our expert team has designed the solutions for RD Sharma Solutions for Class 12 Maths Exercise 4.14 Chapter 4 Inverse Trigonometric Functions to help students prepare for their exams with ease. RD Sharma Solutions for Class 12 is one of the best reference materials for CBSE students. Learners can download the pdf from the links provided below.

Experts suggest students practice the solutions many number of times to yield good results in their exams. In Exercise 4.14 of Chapter 4 Inverse Trigonometric Functions, we shall discuss the ASA congruence condition for two congruent triangles and their properties.

Download the PDF of RD Sharma Solutions For Class 12 Chapter 4 – Inverse Trigonometric Functions Exercise 4.14

 

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Access answers to Maths RD Sharma Solutions For Class 12 Chapter 4 – Inverse Trigonometric Functions Exercise 4.14

Exercise 4.14 Page No: 4.115

1. Evaluate the following:

(i) tan {2 tan-1 (1/5) – π/4}

(ii) Tan {1/2 sin-1 (3/4)}

(iii) Sin {1/2 cos-1 (4/5)}

(iv) Sin (2 tan -1 2/3) + cos (tan-1 √3)

Solution:

(i) Given tan {2 tan-1 (1/5) – π/4}

RD Sharma Solutions for Class 12 Maths Chapter 4 Inverse Trigonometric Functions Image 68

RD Sharma Solutions for Class 12 Maths Chapter 4 Inverse Trigonometric Functions Image 69

(ii) Given tan {1/2 sin-1 (3/4)}

RD Sharma Solutions for Class 12 Maths Chapter 4 Inverse Trigonometric Functions Image 70

RD Sharma Solutions for Class 12 Maths Chapter 4 Inverse Trigonometric Functions Image 71

RD Sharma Solutions for Class 12 Maths Chapter 4 Inverse Trigonometric Functions Image 72

(iii) Given sin {1/2 cos-1 (4/5)}

RD Sharma Solutions for Class 12 Maths Chapter 4 Inverse Trigonometric Functions Image 73

RD Sharma Solutions for Class 12 Maths Chapter 4 Inverse Trigonometric Functions Image 74

(iv) Given Sin (2 tan -1 2/3) + cos (tan-1 √3)

RD Sharma Solutions for Class 12 Maths Chapter 4 Inverse Trigonometric Functions Image 75

RD Sharma Solutions for Class 12 Maths Chapter 4 Inverse Trigonometric Functions Image 76

2. Prove the following results:

(i) 2 sin-1 (3/5) = tan-1 (24/7)

(ii) tan-1 ¼ + tan-1 (2/9) = ½ cos-1 (3/5) = ½ sin-1 (4/5)

(iii) tan-1 (2/3) = ½ tan-1 (12/5)

(iv) tan-1 (1/7) + 2 tan-1 (1/3) = π/4

(v) sin-1 (4/5) + 2 tan-1 (1/3) = π/2

(vi) 2 sin-1 (3/5) – tan-1 (17/31) = π/4

(vii) 2 tan-1 (1/5) + tan-1 (1/8) = tan-1 (4/7)

(viii) 2 tan-1 (3/4) – tan-1 (17/31) = π/4

(ix) 2 tan-1 (1/2) + tan-1 (1/7) = tan-1 (31/17)

(x) 4 tan-1(1/5) – tan-1(1/239) = π/4

Solution:

(i) Given 2 sin-1 (3/5) = tan-1 (24/7)

RD Sharma Solutions for Class 12 Maths Chapter 4 Inverse Trigonometric Functions Image 77

RD Sharma Solutions for Class 12 Maths Chapter 4 Inverse Trigonometric Functions Image 78

Hence, proved.

(ii) Given tan-1 ¼ + tan-1 (2/9) = ½ cos-1 (3/5) = ½ sin-1 (4/5)

RD Sharma Solutions for Class 12 Maths Chapter 4 Inverse Trigonometric Functions Image 79

RD Sharma Solutions for Class 12 Maths Chapter 4 Inverse Trigonometric Functions Image 80

RD Sharma Solutions for Class 12 Maths Chapter 4 Inverse Trigonometric Functions Image 81

RD Sharma Solutions for Class 12 Maths Chapter 4 Inverse Trigonometric Functions Image 82

Hence, proved.

(iii) Given tan-1 (2/3) = ½ tan-1 (12/5)

RD Sharma Solutions for Class 12 Maths Chapter 4 Inverse Trigonometric Functions Image 83

RD Sharma Solutions for Class 12 Maths Chapter 4 Inverse Trigonometric Functions Image 84

Hence, proved.

(iv) Given tan-1 (1/7) + 2 tan-1 (1/3) = π/4

RD Sharma Solutions for Class 12 Maths Chapter 4 Inverse Trigonometric Functions Image 85

RD Sharma Solutions for Class 12 Maths Chapter 4 Inverse Trigonometric Functions Image 86

Hence, proved.

(v) Given sin-1 (4/5) + 2 tan-1 (1/3) = π/2

RD Sharma Solutions for Class 12 Maths Chapter 4 Inverse Trigonometric Functions Image 87

RD Sharma Solutions for Class 12 Maths Chapter 4 Inverse Trigonometric Functions Image 88

RD Sharma Solutions for Class 12 Maths Chapter 4 Inverse Trigonometric Functions Image 89

RD Sharma Solutions for Class 12 Maths Chapter 4 Inverse Trigonometric Functions Image 90

(vi) Given 2 sin-1 (3/5) – tan-1 (17/31) = π/4

RD Sharma Solutions for Class 12 Maths Chapter 4 Inverse Trigonometric Functions Image 91

RD Sharma Solutions for Class 12 Maths Chapter 4 Inverse Trigonometric Functions Image 92

RD Sharma Solutions for Class 12 Maths Chapter 4 Inverse Trigonometric Functions Image 93

(vii) Given 2 tan-1 (1/5) + tan-1 (1/8) = tan-1 (4/7)

RD Sharma Solutions for Class 12 Maths Chapter 4 Inverse Trigonometric Functions Image 94

RD Sharma Solutions for Class 12 Maths Chapter 4 Inverse Trigonometric Functions Image 95

RD Sharma Solutions for Class 12 Maths Chapter 4 Inverse Trigonometric Functions Image 96

Hence, proved.

(viii) Given 2 tan-1 (3/4) – tan-1 (17/31) = π/4

RD Sharma Solutions for Class 12 Maths Chapter 4 Inverse Trigonometric Functions Image 97

RD Sharma Solutions for Class 12 Maths Chapter 4 Inverse Trigonometric Functions Image 98

RD Sharma Solutions for Class 12 Maths Chapter 4 Inverse Trigonometric Functions Image 99

Hence, proved.

(ix) Given 2 tan-1 (1/2) + tan-1 (1/7) = tan-1 (31/17)

RD Sharma Solutions for Class 12 Maths Chapter 4 Inverse Trigonometric Functions Image 100

RD Sharma Solutions for Class 12 Maths Chapter 4 Inverse Trigonometric Functions Image 101

Hence, proved.

(x) Given 4 tan-1(1/5) – tan-1(1/239) = π/4

RD Sharma Solutions for Class 12 Maths Chapter 4 Inverse Trigonometric Functions Image 102

RD Sharma Solutions for Class 12 Maths Chapter 4 Inverse Trigonometric Functions Image 103

Hence, proved.

3. If sin-1 (2a/1 + a2) – cos-1(1 – b2/1 + b2) = tan-1(2x/1 – x2), then prove that x = (a – b)/ (1 + a b)

Solution:

Given sin-1 (2a/1 + a2) – cos-1(1 – b2/1 + b2) = tan-1(2x/1 – x2)

RD Sharma Solutions for Class 12 Maths Chapter 4 Inverse Trigonometric Functions Image 104

RD Sharma Solutions for Class 12 Maths Chapter 4 Inverse Trigonometric Functions Image 105

Hence, proved.

4. Prove that:

(i) tan-1{(1 – x2)/ 2x)} + cot-1{(1 – x2)/ 2x)} = π/2

(ii) sin {tan-1 (1 – x2)/ 2x) + cos-1 (1 – x2)/ (1 + x2)} = 1

Solution:

(i) Given tan-1{(1 – x2)/ 2x)} + cot-1{(1 – x2)/ 2x)} = π/2

RD Sharma Solutions for Class 12 Maths Chapter 4 Inverse Trigonometric Functions Image 106

RD Sharma Solutions for Class 12 Maths Chapter 4 Inverse Trigonometric Functions Image 107

Hence, proved.

(ii) Given sin {tan-1 (1 – x2)/ 2x) + cos-1 (1 – x2)/ (1 + x2)}

RD Sharma Solutions for Class 12 Maths Chapter 4 Inverse Trigonometric Functions Image 108

RD Sharma Solutions for Class 12 Maths Chapter 4 Inverse Trigonometric Functions Image 109

RD Sharma Solutions for Class 12 Maths Chapter 4 Inverse Trigonometric Functions Image 110

RD Sharma Solutions for Class 12 Maths Chapter 4 Inverse Trigonometric Functions Image 111

Hence, proved.

5. If sin-1 (2a/ 1+ a2) + sin-1 (2b/ 1+ b2) = 2 tan-1 x, prove that x = (a + b/ 1 – a b)

Solution:

Given sin-1 (2a/ 1+ a2) + sin-1 (2b/ 1+ b2) = 2 tan-1 x

RD Sharma Solutions for Class 12 Maths Chapter 4 Inverse Trigonometric Functions Image 112
RD Sharma Solutions for Class 12 Maths Chapter 4 Inverse Trigonometric Functions Image 113

Hence, proved.


Access other exercises of RD Sharma Solutions For Class 12 Chapter 4 – Inverse Trigonometric Functions

Exercise 4.1 Solutions

Exercise 4.2 Solutions

Exercise 4.3 Solutions

Exercise 4.4 Solutions

Exercise 4.5 Solutions

Exercise 4.6 Solutions

Exercise 4.7 Solutions

Exercise 4.8 Solutions

Exercise 4.9 Solutions

Exercise 4.10 Solutions

Exercise 4.11 Solutions

Exercise 4.12 Solutions

Exercise 4.13 Solutions

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