This document contains examples of calculating principal values of inverse trigonometric functions. It includes solving equations involving inverse trigonometric functions and proving trigonometric identities related to inverse functions. The document provides step-by-step workings for 17 questions involving inverse sine, cosine, tangent, cotangent, secant and cosecant functions, demonstrating how to determine principal values from given ranges and simplify expressions involving inverse trig functions.

1. Class XII Chapter 2 – Inverse Trigonometric Functions Maths
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Exercise 2.1
Question 1:
Find the principal value of
Answer
Let sin-1
Then sin y =
We know that the range of the principal value branch of sin−1
is
and sin
Therefore, the principal value of
Question 2:
Find the principal value of
Answer
We know that the range of the principal value branch of cos−1
is
.
Therefore, the principal value of .
Question 3:
Find the principal value of cosec−1
(2)
Answer

2. Class XII Chapter 2 – Inverse Trigonometric Functions Maths
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Let cosec−1
(2) = y. Then,
We know that the range of the principal value branch of cosec−1
is
Therefore, the principal value of
Question 4:
Find the principal value of
Answer
We know that the range of the principal value branch of tan−1
is
Therefore, the principal value of
Question 5:
Find the principal value of
Answer
We know that the range of the principal value branch of cos−1
is
Therefore, the principal value of

3. Class XII Chapter 2 – Inverse Trigonometric Functions Maths
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Question 6:
Find the principal value of tan−1
(−1)
Answer
Let tan−1
(−1) = y. Then,
We know that the range of the principal value branch of tan−1
is
Therefore, the principal value of
Question 7:
Find the principal value of
Answer
We know that the range of the principal value branch of sec−1
is
Therefore, the principal value of
Question 8:
Find the principal value of
Answer

4. Class XII Chapter 2 – Inverse Trigonometric Functions Maths
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We know that the range of the principal value branch of cot−1
is (0,π) and
Therefore, the principal value of
Question 9:
Find the principal value of
Answer
We know that the range of the principal value branch of cos−1
is [0,π] and
.
Therefore, the principal value of
Question 10:
Find the principal value of
Answer
We know that the range of the principal value branch of cosec−1
is
Therefore, the principal value of

5. Class XII Chapter 2 – Inverse Trigonometric Functions Maths
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Question 11:
Find the value of
Answer
Question 12:
Find the value of
Answer

6. Class XII Chapter 2 – Inverse Trigonometric Functions Maths
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Question 13:
Find the value of if sin−1
x = y, then
(A) (B)
(C) (D)
Answer
It is given that sin−1
x = y.
We know that the range of the principal value branch of sin−1
is
Therefore, .
Question 14:
Find the value of is equal to
(A) π (B) (C) (D)
Answer

7. Class XII Chapter 2 – Inverse Trigonometric Functions Maths
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8. Class XII Chapter 2 – Inverse Trigonometric Functions Maths
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Exercise 2.2
Question 1:
Prove
Answer
To prove:
Let x = sinθ. Then,
We have,
R.H.S. =
= 3θ
= L.H.S.
Question 2:
Prove
Answer
To prove:
Let x = cosθ. Then, cos−1
x =θ.
We have,

9. Class XII Chapter 2 – Inverse Trigonometric Functions Maths
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Question 3:
Prove
Answer
To prove:
Question 4:
Prove
Answer
To prove:

10. Class XII Chapter 2 – Inverse Trigonometric Functions Maths
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Question 5:
Write the function in the simplest form:
Answer

11. Class XII Chapter 2 – Inverse Trigonometric Functions Maths
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Question 6:
Write the function in the simplest form:
Answer
Put x = cosec θ ⇒ θ = cosec−1
x
Question 7:
Write the function in the simplest form:

12. Class XII Chapter 2 – Inverse Trigonometric Functions Maths
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Answer
Question 8:
Write the function in the simplest form:
Answer

13. Class XII Chapter 2 – Inverse Trigonometric Functions Maths
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Question 9:
Write the function in the simplest form:
Answer
Question 10:
Write the function in the simplest form:
Answer

14. Class XII Chapter 2 – Inverse Trigonometric Functions Maths
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Question 11:
Find the value of
Answer
Let . Then,
Question 12:

15. Class XII Chapter 2 – Inverse Trigonometric Functions Maths
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Find the value of
Answer
Question 13:
Find the value of
Answer
Let x = tan θ. Then, θ = tan−1
x.
Let y = tan Φ. Then, Φ = tan−1
y.
Question 14:

16. Class XII Chapter 2 – Inverse Trigonometric Functions Maths
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If , then find the value of x.
Answer
On squaring both sides, we get:

17. Class XII Chapter 2 – Inverse Trigonometric Functions Maths
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Hence, the value of x is
Question 15:
If , then find the value of x.
Answer

18. Class XII Chapter 2 – Inverse Trigonometric Functions Maths
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Hence, the value of x is
Question 16:
Find the values of
Answer

19. Class XII Chapter 2 – Inverse Trigonometric Functions Maths
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We know that sin−1
(sin x) = x if , which is the principal value branch of
sin−1
x.
Here,
Now, can be written as:
Question 17:
Find the values of
Answer
We know that tan−1
(tan x) = x if , which is the principal value branch of
tan−1
x.
Here,
Now, can be written as:

20. Class XII Chapter 2 – Inverse Trigonometric Functions Maths
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Question 18:
Find the values of
Answer
Let . Then,
Question 19:

21. Class XII Chapter 2 – Inverse Trigonometric Functions Maths
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Find the values of is equal to
(A) (B) (C) (D)
Answer
We know that cos−1
(cos x) = x if , which is the principal value branch of cos
−1
x.
Here,
Now, can be written as:
The correct answer is B.
Question 20:
Find the values of is equal to
(A) (B) (C) (D) 1
Answer
Let . Then,
We know that the range of the principal value branch of .

22. Class XII Chapter 2 – Inverse Trigonometric Functions Maths
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∴
The correct answer is D.

23. Class XII Chapter 2 – Inverse Trigonometric Functions Maths
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Miscellaneous Solutions
Question 1:
Find the value of
Answer
We know that cos−1
(cos x) = x if , which is the principal value branch of cos
−1
x.
Here,
Now, can be written as:
Question 2:
Find the value of
Answer
We know that tan−1
(tan x) = x if , which is the principal value branch of
tan −1
x.
Here,
Now, can be written as:

24. Class XII Chapter 2 – Inverse Trigonometric Functions Maths
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Question 3:
Prove
Answer
Now, we have:

25. Class XII Chapter 2 – Inverse Trigonometric Functions Maths
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Question 4:
Prove
Answer
Now, we have:

26. Class XII Chapter 2 – Inverse Trigonometric Functions Maths
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Question 5:
Prove
Answer

27. Class XII Chapter 2 – Inverse Trigonometric Functions Maths
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Now, we will prove that:

28. Class XII Chapter 2 – Inverse Trigonometric Functions Maths
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Question 6:
Prove
Answer
Now, we have:

29. Class XII Chapter 2 – Inverse Trigonometric Functions Maths
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Question 7:
Prove
Answer
Using (1) and (2), we have

30. Class XII Chapter 2 – Inverse Trigonometric Functions Maths
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Question 8:
Prove
Answer

31. Class XII Chapter 2 – Inverse Trigonometric Functions Maths
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Question 9:
Prove
Answer

32. Class XII Chapter 2 – Inverse Trigonometric Functions Maths
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Question 10:
Prove
Answer
Question 11:
Prove [Hint: putx = cos 2θ]
Answer

33. Class XII Chapter 2 – Inverse Trigonometric Functions Maths
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Question 12:
Prove
Answer

34. Class XII Chapter 2 – Inverse Trigonometric Functions Maths
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Question 13:
Solve
Answer
Question 14:
Solve
Answer

35. Class XII Chapter 2 – Inverse Trigonometric Functions Maths
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Question 15:
Solve is equal to
(A) (B) (C) (D)
Answer
Let tan−1
x = y. Then,
The correct answer is D.
Question 16:
Solve , then x is equal to

36. Class XII Chapter 2 – Inverse Trigonometric Functions Maths
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(A) (B) (C) 0 (D)
Answer
Therefore, from equation (1), we have
Put x = sin y. Then, we have:
But, when , it can be observed that:

37. Class XII Chapter 2 – Inverse Trigonometric Functions Maths
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is not the solution of the given equation.
Thus, x = 0.
Hence, the correct answer is C.
Question 17:
Solve is equal to
(A) (B). (C) (D)
Answer

38. Class XII Chapter 2 – Inverse Trigonometric Functions Maths
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Hence, the correct answer is C.