Which among the following ratio, is not opposite of each other?

Options

A. `sintheta and costheta`
B. `costheta and sectheta`
C. `tantheta and cottheta`
D. `cosectheta and sintheta`

Answer

To solve the question "Which among the following ratio is not opposite to each other?", we will analyze the given options step by step.

### Step 1: Understand the Concept of Opposite Ratios
Two ratios are considered opposite if the product of the two ratios equals 1. For example, if we have a ratio \( x \), its opposite would be \( \frac{1}{x} \). Therefore, if \( x \cdot \frac{1}{x} = 1 \), they are opposite.

### Step 2: Analyze Each Option
Let's denote the options as follows for clarity:
- Option A: \( \sin \theta \) and \( \cos \theta \)
- Option B: \( \cos \theta \) and \( \sec \theta \)
- Option C: \( \tan \theta \) and \( \cot \theta \)
- Option D: \( \sin \theta \) and \( \csc \theta \)

We will check each pair to see if they are opposite.

### Step 3: Check Option A
- Ratios: \( \sin \theta \) and \( \cos \theta \)
- Product: \( \sin \theta \cdot \cos \theta \)
- This does not equal 1, so they are not opposite.

### Step 4: Check Option B
- Ratios: \( \cos \theta \) and \( \sec \theta \)
- Recall that \( \sec \theta = \frac{1}{\cos \theta} \)
- Product: \( \cos \theta \cdot \sec \theta = \cos \theta \cdot \frac{1}{\cos \theta} = 1 \)
- Thus, they are opposite.

### Step 5: Check Option C
- Ratios: \( \tan \theta \) and \( \cot \theta \)
- Recall that \( \cot \theta = \frac{1}{\tan \theta} \)
- Product: \( \tan \theta \cdot \cot \theta = \tan \theta \cdot \frac{1}{\tan \theta} = 1 \)
- Thus, they are opposite.

### Step 6: Check Option D
- Ratios: \( \sin \theta \) and \( \csc \theta \)
- Recall that \( \csc \theta = \frac{1}{\sin \theta} \)
- Product: \( \sin \theta \cdot \csc \theta = \sin \theta \cdot \frac{1}{\sin \theta} = 1 \)
- Thus, they are opposite.

### Conclusion
From our analysis, we found that:
- Option A: \( \sin \theta \) and \( \cos \theta \) are **not** opposite.
- Options B, C, and D are all opposite.

Therefore, the answer is **Option A**.