What is the value of `(sectheta)/(tantheta+cottheta)`?

Options

A. `costheta`
B. `tantheta`
C. `sintheta`
D. `cottheta`

Answer

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

### Step 1: Write down the definitions of the trigonometric functions
- \(\sec \theta = \frac{1}{\cos \theta}\)
- \(\tan \theta = \frac{\sin \theta}{\cos \theta}\)
- \(\cot \theta = \frac{\cos \theta}{\sin \theta}\)

### Step 2: Substitute the definitions into the expression
Substituting the definitions into the expression gives us:
\[ \frac{\sec \theta}{\tan \theta + \cot \theta} = \frac{\frac{1}{\cos \theta}}{\frac{\sin \theta}{\cos \theta} + \frac{\cos \theta}{\sin \theta}} \]

### Step 3: Simplify the denominator
To simplify the denominator \(\tan \theta + \cot \theta\):
\[ \tan \theta + \cot \theta = \frac{\sin \theta}{\cos \theta} + \frac{\cos \theta}{\sin \theta} \]
Finding a common denominator (which is \(\sin \theta \cos \theta\)):
\[ = \frac{\sin^2 \theta + \cos^2 \theta}{\sin \theta \cos \theta} \]

### Step 4: Use the Pythagorean identity
Using the identity \(\sin^2 \theta + \cos^2 \theta = 1\):
\[ = \frac{1}{\sin \theta \cos \theta} \]

### Step 5: Substitute back into the expression
Now substituting back into our original expression:
\[ \frac{\frac{1}{\cos \theta}}{\frac{1}{\sin \theta \cos \theta}} = \frac{1}{\cos \theta} \cdot \frac{\sin \theta \cos \theta}{1} = \frac{\sin \theta}{1} = \sin \theta \]

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

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