What is the value of `(sec theta)/(tan theta +cot theta)` ?

Options

A. `cos theta`
B. `tan theta`
C. `sin theta`
D. `cot theta`

Answer

To solve the expression \(\frac{\sec \theta}{\tan \theta + \cot \theta}\), we will follow these steps:

### Step 1: Rewrite the trigonometric functions
We start by rewriting \(\sec \theta\), \(\tan \theta\), and \(\cot \theta\) in terms of sine and cosine:
- \(\sec \theta = \frac{1}{\cos \theta}\)
- \(\tan \theta = \frac{\sin \theta}{\cos \theta}\)
- \(\cot \theta = \frac{\cos \theta}{\sin \theta}\)

### Step 2: Substitute these values into the expression
Now, we substitute these values into the expression:
\[ \frac{\sec \theta}{\tan \theta + \cot \theta} = \frac{\frac{1}{\cos \theta}}{\frac{\sin \theta}{\cos \theta} + \frac{\cos \theta}{\sin \theta}} \]

### Step 3: Simplify the denominator
Next, we simplify the denominator:
\[ \tan \theta + \cot \theta = \frac{\sin \theta}{\cos \theta} + \frac{\cos \theta}{\sin \theta} \]
To combine these fractions, we find a common denominator, which is \(\sin \theta \cos \theta\):
\[ = \frac{\sin^2 \theta + \cos^2 \theta}{\sin \theta \cos \theta} \]
Using the Pythagorean identity, \(\sin^2 \theta + \cos^2 \theta = 1\), we can simplify this to:
\[ = \frac{1}{\sin \theta \cos \theta} \]

### Step 4: Substitute back into the expression
Now we substitute this back into our expression:
\[ \frac{\frac{1}{\cos \theta}}{\frac{1}{\sin \theta \cos \theta}} = \frac{1}{\cos \theta} \times \frac{\sin \theta \cos \theta}{1} = \frac{\sin \theta}{1} = \sin \theta \]

### Conclusion
Thus, the value of \(\frac{\sec \theta}{\tan \theta + \cot \theta}\) is:
\[ \sin \theta \]

### Final Answer
The correct answer is \(\sin \theta\).

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