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

Options

A. `1-cos^2""theta`
B. `2sintheta`
C. `cosectheta`
D. `sec^2""theta`

Answer

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

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

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

### Step 2: Combine the terms in the numerator
To combine \(\frac{\cos \theta}{\sin \theta}\) and \(\frac{\sin \theta}{\cos \theta}\), we need a common denominator:
\[ \frac{\cos^2 \theta + \sin^2 \theta}{\sin \theta \cos \theta} \]
Using the Pythagorean identity, we know that \(\cos^2 \theta + \sin^2 \theta = 1\), so we can simplify this to:
\[ \frac{1}{\sin \theta \cos \theta} \]

### Step 3: Substitute back into the expression
Now we substitute this back into our expression:
\[ \frac{\frac{1}{\sin \theta \cos \theta}}{\frac{1}{\cos \theta}} = \frac{1}{\sin \theta \cos \theta} \times \cos \theta = \frac{1}{\sin \theta} \]

### Step 4: Final simplification
The expression \(\frac{1}{\sin \theta}\) is equal to \(\csc \theta\).

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