In which of the following intervals the inequality, `sinx < cos x < tanx < cot x` can hold good ? (a) `((7pi)/4,2pi)` (b) `((3pi)/4,pi)` (c) `((5pi)/4,(3pi)/2)` (d) `(0,(pi)/4)`

Options

A. `((7pi)/4,2pi)`
B. `((3pi)/4,pi)`
C. `((5pi)/4,(3pi)/2)`
D. `(0,(pi)/4)`

Explanation

To solve the inequality `sin x < cos x < tan x < cot x`, we will analyze each part of the inequality in the given intervals.

### Step-by-step Solution:

1. **Understanding the Trigonometric Functions**:
- We know the definitions:
- `tan x = sin x / cos x`
- `cot x = cos x / sin x`
- The inequalities can be rewritten as:
- `sin x < cos x`
- `cos x < sin x / cos x` (which simplifies to `cos^2 x < sin x`)
- `sin x / cos x < cos x / sin x` (which simplifies to `sin^2 x < cos^2 x`)

2. **Analyzing the Intervals**:
- The intervals provided are:
- (a) `((7pi)/4, 2pi)`
- (b) `((3pi)/4, pi)`
- (c) `((5pi)/4, (3pi)/2)`
- (d) `(0, (pi)/4)`

3. **Evaluating Each Interval**:

- **Interval (a) `((7pi)/4, 2pi)`**:
- In this interval, `x` is in the fourth quadrant.
- Here, `sin x < 0` and `cos x > 0`. Therefore, `sin x < cos x` holds true.
- However, `tan x` and `cot x` are both negative and positive respectively, which makes `tan x < cot x` false. Thus, this interval does not satisfy the inequality.

- **Interval (b) `((3pi)/4, pi)`**:
- In this interval, `x` is in the second quadrant.
- Here, `sin x > 0` and `cos x < 0`. Therefore, `sin x < cos x` is false. Thus, this interval does not satisfy the inequality.

- **Interval (c) `((5pi)/4, (3pi)/2)`**:
- In this interval, `x` is in the third quadrant.
- Here, `sin x < 0` and `cos x < 0`. Therefore, `sin x < cos x` is false. Thus, this interval does not satisfy the inequality.

- **Interval (d) `(0, (pi)/4)`**:
- In this interval, `x` is in the first quadrant.
- Here, `sin x > 0` and `cos x > 0`. Therefore, `sin x < cos x` holds true.
- Also, `tan x < cot x` holds true since both are positive in this interval.
- Thus, this interval satisfies all parts of the inequality.

4. **Conclusion**:
- The only interval where the inequality `sin x < cos x < tan x < cot x` holds true is **(d) `(0, (pi)/4)`**.