Which is equal to `sectheta."cosec"theta` ?

Options

A. `sin theta + cos theta`
B. `tan theta + cot theta`
C. `2(tan theta + cot theta)`
D. `2(sintheta + cos theta)`

Answer

To solve the problem of finding what is equal to \( \sec \theta \cdot \csc \theta \), we can follow these steps:

### Step 1: Write the definitions of secant and cosecant
The secant and cosecant functions are defined as follows:
- \( \sec \theta = \frac{1}{\cos \theta} \)
- \( \csc \theta = \frac{1}{\sin \theta} \)

### Step 2: Substitute the definitions into the expression
Substituting the definitions into the expression \( \sec \theta \cdot \csc \theta \):
\[ \sec \theta \cdot \csc \theta = \frac{1}{\cos \theta} \cdot \frac{1}{\sin \theta} \]

### Step 3: Combine the fractions
Now, we can combine the fractions:
\[ \sec \theta \cdot \csc \theta = \frac{1}{\cos \theta \cdot \sin \theta} \]

### Step 4: Use the identity for sine and cosine
We know that:
\[ \sin \theta \cdot \cos \theta = \frac{1}{2} \sin(2\theta) \]
However, we don't need to use this identity directly for our options. Instead, we can express the product in terms of tangent and cotangent.

### Step 5: Rewrite using tangent and cotangent
We can rewrite the expression as:
\[ \sec \theta \cdot \csc \theta = \frac{1}{\sin \theta \cdot \cos \theta} = \frac{2}{\sin(2\theta)} \]
But for the purpose of matching with the options, we can also express it as:
\[ \sec \theta \cdot \csc \theta = \tan \theta + \cot \theta \]

### Step 6: Check the options
Now, we can check which of the options given matches:
1. \( \sin \theta + \cos \theta \)
2. \( \tan \theta + \cot \theta \)
3. \( 2 \tan \theta + \cot \theta \)
4. \( 2 \sin \theta + \cos \theta \)

From our derivation, we see that \( \sec \theta \cdot \csc \theta = \tan \theta + \cot \theta \), which corresponds to option 2.

### Conclusion
Thus, the answer is:
\[ \sec \theta \cdot \csc \theta = \tan \theta + \cot \theta \]

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