RBSE Solutions for Class 12 Maths Chapter 2 Inverse Circular Functions Ex 2.1

Rajasthan Board RBSE Class 12 Maths Chapter 2 Inverse Circular Functions Ex 2.1

Question 1.
Find the principal value of the following angles:

Solution:
(i) sin^-1(1)

Prove the following : (Q. 2 to 8)
Question 2.

Solution:

Question 3.

Solution:

Question 4.

Solution:

Question 5.
sec² (tan^-1 2) + cosec² (cot^-1 3) = 15
Solution:
Let tan^-12 = θ
⇒ tan θ = 2
sec² θ = 1 + tan² θ
= 1 + (2)²
= 1 + 4 = 5
∴ sec² (tan^-12) = 5
Let cot^-13 = Φ = cot Φ = 3
∴ cosec² Φ = 1 + cot² Φ
= 1 + (3)² = 1 + 9 = 10
∴ cosec²(cot^-1 3) = 10 …..(ii)
Adding eq. (i) and (ii), we get
sec² (tan^-12) + cosec² (cot^-13) = 5 + 10
⇒ sec² (tan^-12) + cosec²(cot^-13) = 15
Hence proved.

Question 6.

Solution:

Question 7.

Solution:

Question 8.

Solution:

Question 9.
If cos^-1x + cos^-1y + cos^-1z = π, then prove that x² + y² + x² + 2xyz = 1. Solution:
cos^-1x + cos^-1y + cos^-1z = π
(According to question)
cos^-1x + cos^-1 y = π – cos^-1z

Question 10.
If sin^-1x + sin^-1 y + sin^-1z = π, then prove that:

Solution:
Let sin^-1 x = A ⇒ x = sin A
sin^-1 y = B ⇒ y = sin B
sin^-1 z = C ⇒ z = sin C
∵ sin^-1 x + sin^-1 y + sin^-1 z = π
∴ A + B + C = π
∴ To prove

Question 11.
If tan^-1x + tan^-1y + tan^-1z = π/2, then prove that xy + yz + zx = 1.
Solution:
According to question,

Question 12.
If

then prove that x + y + z = xyz.
Solution:
Let x = tan A, y = tan B, z = tan C

Question 13.
If

then prove that x + y + z = xyz.
Solution:
According to question

Question 14.
Prove that :
tan^-1x + cot^-1(x + 1) = tan^-1(x² + x + 1).
Solution:
L.H.S. = tan^-1x + cot^-1(x + 1)

Question 15.
If tan^-1x, tan^-1y, tan^-1z are in arithmetic progression, then prove that
y²(x +z) + 2y(1 – xz) – x – z = 0.
Solution:
tan^-1x, tan^-1y, tan^-1z are in A.P.

Question 16.
If α, β, γ be the roots of equation x³ + px² + qx + p = 0, then prove that, except a special condition,
tan^-1α + tan^-1 β + tan^-1γ = nπ also find the special condition, when it does so on.
Solution:
Given, α, β, γ are root of equation x³+ px² + qx + p = 0


Special Situation: When sum and product of roots is not equal, then tan^-1 α + tan^-1 β + tan^-1γ ≠ nπ

Solve the following euqations : (Q. 17 to 25)

Question 17.

Solution:

Question 18.

Solution:

Question 19.

Solution:

Question 20.

Solution:

Question 21.

Solution:

Question 22.

Solution:

Question 23.
sin 2 (cos^-1 {cot (2 tan^-1x)} = 0
Solution:
Given equation

Question 24.

Solution:

Question 25.

Solution:
According to question,

RBSE Solutions for Class 12 Maths