## 1. Which of the following is the principal value branch of sin⁻¹x?

### Answer:

### Explanation:

The principal value branch of sin⁻¹x is [-π/2, π/2].

## 2. If sin⁻¹x + sin⁻¹y = π/2, then y is equal to:

### Answer:

### Explanation:

Using the property sin⁻¹x + cos⁻¹x = π/2, we get y = cos x.

## 3. The value of tan(sin⁻¹(3/5)) is:

### Answer:

### Explanation:

Let sin⁻¹(3/5) = θ. Then, sin θ = 3/5 and cos θ = 4/5. tan θ = sin θ/cos θ = 4/3.

## 4. The principal value of cos⁻¹(-1/2) is:

### Answer:

### Explanation:

The principal value of cos⁻¹x lies in [0, π]. Therefore, for x = -1/2, it is 2π/3.

## 5. The domain of the function tan⁻¹x is:

### Answer:

### Explanation:

The function tan⁻¹x is defined for all real values.

## 6. The range of the function sec⁻¹x is:

### Answer:

### Explanation:

The function sec⁻¹x takes values in [0, π] but is not defined at π/2.

## 7. The value of sin⁻¹(0) is:

### Answer:

### Explanation:

The inverse sine of 0 is 0.

## 8. If x = tan⁻¹(√3), then the value of sin(2x) is:

### Answer:

### Explanation:

x = π/3. Using the double angle formula, sin(2x) = sin(2π/3) = 0.

## 9. The value of cos⁻¹(0) is:

### Answer:

### Explanation:

The inverse cosine of 0 is π/2.

## 10. Which of the following is not defined?

### Answer:

### Explanation:

The function cos⁻¹x is defined for x in [-1, 1].

## 11. The value of sin(tan⁻¹(1)) is:

### Answer:

### Explanation:

Let θ = tan⁻¹(1). Then, tan θ = 1 implies θ = π/4. Thus, sin(π/4) = √2/2.

## 12. The principal value of cot⁻¹(1) is:

### Answer:

### Explanation:

The principal value of cot⁻¹(1) is π/4.

## 13. The value of tan⁻¹(1) + cot⁻¹(1) is:

### Answer:

### Explanation:

tan⁻¹(1) is π/4 and cot⁻¹(1) is π/4. Their sum is π.

## 14. The domain of cos⁻¹x is:

### Answer:

### Explanation:

The function cos⁻¹x is defined for x in [-1, 1].

## 15. The range of sin⁻¹x is:

### Answer:

### Explanation:

The function sin⁻¹x has its values in [-π/2, π/2].

## 16. If sin(2x) = 1, then x is:

### Answer:

### Explanation:

2x = π/2. Thus, x = π/4 = sin⁻¹(1/2).

## 17. The value of cos(sin⁻¹(1/2)) is:

### Answer:

### Explanation:

Let θ = sin⁻¹(1/2). Then, sin θ = 1/2 and cos θ = √3/2.

## 18. The principal value of sec⁻¹(-1) is:

### Answer:

### Explanation:

The principal value of sec⁻¹(-1) is π.

## 19. Which of the following has the range of [0, π]?

### Answer:

### Explanation:

The function cos⁻¹x takes values in [0, π].

## 20. If x = sin⁻¹(1/√2), then x is equal to:

### Answer:

### Explanation:

The inverse sine of 1/√2 is π/4.