Find the values of the following expression. `tantheta+cottheta` is

Options

A. 1
B. `tantheta`
C. `cosectheta cottheta`
D. `sectheta cosectheta`

Answer

To find the value of the expression \( \tan \theta + \cot \theta \), we can follow these steps:

### Step 1: Write the definitions of \( \tan \theta \) and \( \cot \theta \)
We know that:
\[ \tan \theta = \frac{\sin \theta}{\cos \theta} \]
\[ \cot \theta = \frac{\cos \theta}{\sin \theta} \]

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

### Step 3: Find a common denominator
To combine the two fractions, we need a common denominator, which will be \( \sin \theta \cos \theta \):
\[ \tan \theta + \cot \theta = \frac{\sin^2 \theta}{\sin \theta \cos \theta} + \frac{\cos^2 \theta}{\sin \theta \cos \theta} \]

### Step 4: Combine the fractions
Now we can combine the fractions:
\[ \tan \theta + \cot \theta = \frac{\sin^2 \theta + \cos^2 \theta}{\sin \theta \cos \theta} \]

### Step 5: Use the Pythagorean identity
We know from the Pythagorean identity that:
\[ \sin^2 \theta + \cos^2 \theta = 1 \]
So we can substitute this into our expression:
\[ \tan \theta + \cot \theta = \frac{1}{\sin \theta \cos \theta} \]

### Step 6: Rewrite using trigonometric identities
We can rewrite \( \frac{1}{\sin \theta \cos \theta} \) using the identities for cosecant and secant:
\[ \tan \theta + \cot \theta = \frac{1}{\sin \theta \cos \theta} = \frac{2}{\sin(2\theta)} \]

### Final Result
Thus, the final expression for \( \tan \theta + \cot \theta \) is:
\[ \tan \theta + \cot \theta = \frac{1}{\sin \theta \cos \theta} \]