## 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.

sec^{2} (tan^{-1} 2) + cosec^{2} (cot^{-1} 3) = 15

Solution:

Let tan^{-1}2 = θ

⇒ tan θ = 2

sec^{2} θ = 1 + tan^{2} θ

= 1 + (2)^{2}

= 1 + 4 = 5

∴ sec^{2} (tan^{-1}2) = 5

Let cot^{-1}3 = Φ = cot Φ = 3

∴ cosec^{2} Φ = 1 + cot^{2} Φ

= 1 + (3)^{2} = 1 + 9 = 10

∴ cosec^{2}(cot^{-1} 3) = 10 …..(ii)

Adding eq. (i) and (ii), we get

sec^{2} (tan^{-1}2) + cosec^{2} (cot^{-1}3) = 5 + 10

⇒ sec^{2} (tan^{-1}2) + cosec^{2}(cot^{-1}3) = 15

Hence proved.

Question 6.

Solution:

Question 7.

Solution:

Question 8.

Solution:

Question 9.

If cos^{-1}x + cos^{-1}y + cos^{-1}z = π, then prove that x^{2} + y^{2} + x^{2} + 2xyz = 1. Solution:

cos^{-1}x + cos^{-1}y + cos^{-1}z = π

(According to question)

cos^{-1}x + cos^{-1} y = π – cos^{-1}z

Question 10.

If sin^{-1}x + sin^{-1} y + sin^{-1}z = 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^{-1}x + tan^{-1}y + tan^{-1}z = 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^{-1}x + cot^{-1}(x + 1) = tan^{-1}(x^{2} + x + 1).

Solution:

L.H.S. = tan^{-1}x + cot^{-1}(x + 1)

Question 15.

If tan^{-1}x, tan^{-1}y, tan^{-1}z are in arithmetic progression, then prove that

y^{2}(x +z) + 2y(1 – xz) – x – z = 0.

Solution:

tan^{-1}x, tan^{-1}y, tan^{-1}z are in A.P.

Question 16.

If α, β, γ be the roots of equation x^{3} + px^{2} + qx + p = 0, then prove that, except a special condition,

tan^{-1}α + tan^{-1} β + tan^{-1}γ = nπ also find the special condition, when it does so on.

Solution:

Given, α, β, γ are root of equation x^{3}+ px^{2} + qx + p = 0

Special Situation: When sum and product of roots is not equal, then tan^{-1} α + tan^{-1} β + tan^{-1}γ ≠ 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^{-1}x)} = 0

Solution:

Given equation

Question 24.

Solution:

Question 25.

Solution:

According to question,

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