Chapter 2: Inverse Trigonometric Functions
Competency based questions
1
Q.1- If sin (sin−1 2 + cos −1 𝑥) = 1, find the value of x.
Q.2- Write the range of one branch of sin−1 x, other than the principal branch.
1 3 4− √7
Q.3- Show that tan (2 sin−1 4) = .
3
√1+𝑠𝑖𝑛𝑥 + √1−𝑠𝑖𝑛𝑥 𝑥 𝜋
Q.4- Prove that: cot −1 ( ) =2, x𝜖(0, 4 ).
√1+𝑠𝑖𝑛𝑥− √1−𝑠𝑖𝑛𝑥
√1+𝑥 − √1−𝑥 𝜋 1 1
Q.5- - Prove that: tan−1 ( ) =4 - 2 cos −1 𝑥, − ≤x≤1.
√1+𝑥+ √1−𝑥 √2
3 𝑠𝑖𝑛2𝛼 1 𝜋 𝜋
Q.6- Simplify: tan−1 (5+3 𝑐𝑜𝑠2𝛼)+ tan−1 ( 4 tanα ), where - 2 < x < 2 .
𝜋 1 𝑎 𝜋 1 𝑎 2𝑏
Q.7- Prove that: tan (4 + cos −1 𝑏) + tan (4 − cos −1 𝑏 ) = .
2 2 𝑎
5𝜋2
Q.8- If (tan−1 𝑥)2 + (cot −1 𝑥)2 = , find x.
8
Q.9- If sin−1 𝑥 + sin−1 𝑦 + sin−1 𝑧 = 𝜋, then prove that:
x√1 − 𝑥 2 + y√1 − 𝑦 2 + z√1 − 𝑧 2 = 2xyz.
Q.10- If tan−1 𝑎 + tan−1 𝑏 + tan−1 𝑐 = 𝜋, then prove that a + b + c = abc.
𝛼 𝜋 𝛽 𝑠𝑖𝑛𝛼.𝑐𝑜𝑠𝛽
Q.11- Show that: 2 tan−1 {𝑡𝑎𝑛 tan ( − )} = tan−1 ( ).
2 4 2 cosα +𝑠𝑖𝑛𝛽
Q.12- Evaluate the following:
(a) sin−1 ( sin 10) (b) cos −1 ( cos 10) (c) tan−1 ( tan(−6))
𝜋
Q.13- Solve the equation tan−1 √𝑥 2 + 𝑥 + sin−1 √𝑥 2 + 𝑥 + 1 = 2 .
25
Q.14- Simplify: cot (∑23 −1 𝑛
𝑛=1 cot (1 + ∑𝑘=1 2𝑘 )) Ans. 23
Q.15- Find the sum of: csc −1 √5 + csc −1 √65 + csc −1 √325 + …………………∞
𝜋
Ans. 4
Q.16- Find the sum of:
cot −1 ( 2𝑎−1 + 𝑎) + cot −1 ( 2𝑎−1 + 3𝑎) +cot −1 ( 2𝑎−1 + 6𝑎) + cot −1 ( 2𝑎−1 + 10𝑎) +……………………………………∞
𝑎
Ans. cot −1 2
𝑥 𝑦 𝑥2 𝑥𝑦 𝑦2
Q.17- If cos −1 𝑎 + cos −1 𝑏 = 𝛼, prove that - 2𝑎𝑏cos 𝛼 + 𝑏2 = sin2𝛼 .
𝑎2
𝜋
Q.18- If sin−1 (1 − 𝑥) − 2 sin−1 𝑥 = 2 , find the value of x.
9𝜋 9 1 9 2√2
Q.19-Prove that: 8
- 4 sin−1 3 = 4 sin−1 3
.
Q.20- Solve the equation: 2 tan−1 (cos 𝑥) = tan−1 (2 𝑐𝑜𝑠𝑒𝑐 𝑥).
1−𝑥 1
Q.21- Solve the equation: tan−1 (1+𝑥) =2 tan−1 𝑥, where x > 0
