1. CHAPTER 2
2. INVERSE TRIGONOMETRIC
FUNCTIONS
SAY 2020
1
i) If 𝑥 ∈ [−1,1], then sin−1 ⁡𝑥 + cos−1 ⁡𝑥 =
1
ii) If sin⁡[sin−1 ⁡(5) + cos −1 ⁡𝑥] = 1, then find the value of 𝑥.
iii) Find the domain of the function
𝑓(𝑥) = sin−1 ⁡(𝑥 + 1).
MARCH 2020
2
i) If 𝑥𝑦 < 1, tan−1 ⁡𝑥 + tan−1 ⁡𝑦 = ⋯ ….
𝑥−𝑦
a) tan−1 ⁡(
)
1+𝑥𝑦
𝑥+𝑦
b) tan ⁡(1−𝑥𝑦)
tan⁡𝑥+tan⁡𝑦
c) (1−tan⁡𝑥tan⁡𝑦)
tan⁡𝑥−tan⁡𝑦
d) (
)
1+tan⁡𝑥tan⁡𝑦
−1
ii) Solve tan−1 ⁡2𝑥 + tan−1 ⁡3𝑥 =
𝜋
4
SAY 2019
3
3
5
i) If 𝑥 = sin−1 ⁡( ), then which of the following is true?
5
b) 𝑥 = cos−1 ⁡(3)
3
b) 𝑥 = tan−1 ⁡(4)
5
d) 𝑥 = cosec −1 ⁡(4)
3
d) 𝑥 = cot −1 ⁡(4)
3
3
ii) Evaluate tan⁡(sin−1 ⁡5 + cot −1 ⁡2).
4
i) sin−1 ⁡(sin⁡𝑥) = 𝑥 is defined on
𝜋 𝜋
a) 𝑥 ∈ [− 2 , 2 ]
𝜋 𝜋
b) 𝑥 ∈ (− 2 , 2 )
c) 𝑥 ∈ [0, 𝜋]
d) 𝑥 ∈ (0, 𝜋)
13𝜋
)
4
ii) Find the value of sin−1 ⁡(sin⁡
MARCH 2019
5
1−𝑥 2
2𝑥
𝜋
4𝐵 + 2𝐶 = 3 , find the value of 𝑥
SAY 2018
6
12
a) If cos −1 ⁡13 = tan−1 ⁡𝑥, then find 𝑥.
4
12
14
b) Show that cos−1 ⁡5 + cos−1 ⁡13 = tan−1 ⁡33
There was a mistake in the question.
Correct question is:
4
12
56
Show that cos−1 ⁡5 + cos −1 ⁡13 = tan−1 ⁡33 MARCH 2018
7.
2𝑥
If A = sin−1 ⁡(1+𝑥 2 ) , 𝐵 = cos−1 ⁡(1+𝑥2 ) 𝐶 = tan−1 ⁡(1−𝑥 2 ) satisfies the condition 3𝐴 −
a) Identify the function from the above graph.
i) tan−1 ⁡𝑥
ii) sin−1 ⁡𝑥
iii) cos−1 ⁡𝑥
iv) cosec −1 ⁡𝑥
b) Find the domain and range of the function represented in above graph.
1
2
3
c) Prove that tan−1 ⁡2 + tan−1 ⁡11 = tan−1 ⁡4
SAY 2017
8
iii)
𝜋
4
iv)
−𝜋
6
𝜋
a) The principal value of tan−1 ⁡(−√3) is i) 3 ii)
𝜋
−𝜋
3
𝑥
√1+sin⁡𝑥+√1−sin⁡𝑥
)
√1+sin⁡𝑥−√1−sin⁡𝑥
b) If 𝑥 ∈ (0, 2 ), show that cot −1 ⁡(
= 2 MARCH 2017
is i)
𝜋
3
ii)
−𝜋
3
10 a) The principal value of tan−1 ⁡(−√3) is i)
𝜋
3
ii)
−𝜋
3
9
a) The principal value of cot −1 ⁡(−
𝜋
iii) 6 ⁡ iv)
1
)
√3
2𝜋
3
b) Solve:
𝑥−1
𝑥+1
𝜋
tan−1 ⁡(
) + tan−1 ⁡(
)=
𝑥−2
𝑥+2
4
SAY 2016
𝜋
iii) 4 ⁡ iv)
−𝜋
6
1
2
3
b) Show that tan−1 ⁡2 + tan−1 ⁡11 = tan−1 ⁡4
MARCH 2016
11 a) If 𝑥𝑦 < 1, tan−1 ⁡𝑥 + tan−1 ⁡𝑦 =
1
1
31
b) Prove that 2tan−1 ⁡2 + tan−1 ⁡7 = tan−1 ⁡17
SAY 2015
1
12 a) What is the principal value of cos −1 ⁡(− 2)
√1+𝑥 2 −1
),𝑥
𝑥
b) Express tan−1 ⁡(
≠ 0 in the smallest form. MARCH 2015
13 (a) What is the value of sin−1 ⁡(sin⁡160∘ ) ?
(i) 160∘
(ii) 70∘
(iii) −20∘
(iv) 20∘
1
1
31
(b) Prove that 2tan−1 ⁡2 + tan−1 ⁡7 = tan−1 ⁡17
SAY 2014
𝜋
𝜋
𝜋
𝜋
14 a) The principal value of tan−1 ⁡(−1) is ( 4 , − 4 , 𝜋 − 4 , 𝜋 + 4 )
𝑥−1
𝑥+1
𝜋
b) If tan−1 ⁡(𝑥−2) + tan−1 ⁡(𝑥+2) = 4 , then find the value of 𝑥.
MARCH 2014
1
15 a) The principal value of cos−1 ⁡(− 2) is
cos⁡𝑥−sin⁡𝑥
b) Write the function tan−1 ⁡(cos⁡𝑥+sin⁡𝑥) , 0 < 𝑥 < 𝜋 in the simplest form.
SAY 2013
16 a) Show that
1
1
1
1 𝜋
tan−1 ⁡ + tan−1 ⁡ + tan−1 ⁡ + tan−1 ⁡ =
3
5
7
8 4
3cot2 ⁡𝜃−1
3𝑥 2 −1
b) Given that cot⁡3𝜃 = cot3 ⁡𝜃−3cot⁡𝜃, show that cot −1 ⁡(𝑥 3 −3𝑥) , |𝑥| < √3 is 3cot −1 ⁡𝑥
MARCH 2013
1
17 a) Find the principal value of sin−1 ⁡(2) b) Show that
3
8
84
sin−1 ⁡( ) − sin−1 ⁡( ) = cos −1 ⁡( )
5
17
85
3. SAY
1
18 a) If sin⁡(sin−1 ⁡5 + cos−1 ⁡𝑥) = 1, write the value of x.
b) Write the simplest form of
cos⁡𝑥
𝜋
3𝜋
tan−1 ⁡(
),− < 𝑥 <
1 − sin⁡𝑥
2
2
4. MARCH
19 a) The principal value of tan−1 ⁡(1) is
√1+𝑥 2 −1
),𝑥
𝑥
b) Express tan−1 ⁡(
≠ 0 in the smallest form.
SAY 2011
20 a) Given an expression for tan⁡(𝑥 + 𝑦)
b) Prove that for 𝑥𝑦 < 1,
𝑥+𝑦
tan−1 ⁡𝑥 + tan−1 ⁡𝑦 = tan−1 ⁡(
)
1 − 𝑥𝑦
c) Using the above result, prove that
1
1
𝜋
tan−1 ⁡( ) + tan−1 ⁡( ) =
2
3
4
5. MARCH 2011
1
21 a) Find the principal value of cos−1 ⁡(− 2)
cos⁡𝑥
𝜋
𝑥
b) Show that tan−1 ⁡(1−sin⁡𝑥) = 4 + 2 SAY 2010
22 Match the following:
𝐀
𝐁
a) sin−1 ⁡𝑥 + cos−1 ⁡𝑥, 𝑥 ∈ [−1,1]
𝜋
4
b) sin−1 ⁡(sin⁡ 5 )
5𝜋
6
c) cot −1 ⁡(−√3)
𝜋
2
4𝜋
1−√2
1+√2
√
√
d) tan−1 ⁡(1−2 2) + tan−1 ⁡(1+2 2)
𝜋
5