Which one of the following inequality holds good:

Options

A. `cos12^(º) lt tan 12^(º) lt cosec 12^(º) lt cot 12^(º)`
B. `cos 12^(º) lt tan 12^(º) lt cot 12^(º) lt cosec 12^(º)`
C. `tan12^(º) lt cos 12^(º) lt cosec 12^(º) lt cot 12^(º)`
D. `tan 12^(º) lt cos 12^(º) lt cot 12^(º) lt cosec 12^(º)`

Answer

To solve the problem of determining which inequality holds good among cos 12°, tan 12°, cosec 12°, and cot 12°, we can follow these steps:

### Step 1: Understand the Values of Trigonometric Functions
We start by noting the values of the trigonometric functions involved:
- Let \( a = \sin(12^\circ) \)
- Let \( b = \cos(12^\circ) \)

### Step 2: Compare \( \sin(12^\circ) \) and \( \cos(12^\circ) \)
Since \( 12^\circ \) is in the first quadrant (0° to 90°), we know that:
- \( \sin(12^\circ) < \cos(12^\circ) \)
This implies:
- \( a < b \)

### Step 3: Compare \( \csc(12^\circ) \) and \( \cot(12^\circ) \)
Using the definitions of cosecant and cotangent:
- \( \csc(12^\circ) = \frac{1}{\sin(12^\circ)} = \frac{1}{a} \)
- \( \cot(12^\circ) = \frac{\cos(12^\circ)}{\sin(12^\circ)} = \frac{b}{a} \)

Since \( a < b \) (from Step 2), we can conclude:
- \( \csc(12^\circ) = \frac{1}{a} > \frac{b}{a} = \cot(12^\circ) \)
Thus:
- \( \csc(12^\circ) > \cot(12^\circ) \)

### Step 4: Compare \( \cos(12^\circ) \) and \( \tan(12^\circ) \)
Using the definition of tangent:
- \( \tan(12^\circ) = \frac{\sin(12^\circ)}{\cos(12^\circ)} = \frac{a}{b} \)

Since \( a < b \):
- \( \tan(12^\circ) = \frac{a}{b} < 1 \) (because \( a < b \))
This implies:
- \( \cos(12^\circ) = b > \frac{a}{b} = \tan(12^\circ) \)
Thus:
- \( \cos(12^\circ) > \tan(12^\circ) \)

### Step 5: Summarize the Results
From the comparisons, we have:
1. \( \csc(12^\circ) > \cot(12^\circ) \)
2. \( \cos(12^\circ) > \tan(12^\circ) \)

### Conclusion
Based on our analysis, we can conclude that the inequalities that hold good are:
- \( \csc(12^\circ) > \cot(12^\circ) \)
- \( \cos(12^\circ) > \tan(12^\circ) \)

Therefore, the correct answer is that option 4 is valid.

---