`cot x+ tan x=`

Options

A. `cot 2x`
B. `2 cot ^(2) x`
C. `sec x cosec x`
D. `cot ^(2) 2x`

Answer

To solve the equation \( \cot x + \tan x \), we can follow these steps:

### Step 1: Write the definitions of cotangent and tangent
We know that:
\[ \cot x = \frac{\cos x}{\sin x} \]
\[ \tan x = \frac{\sin x}{\cos x} \]

### Step 2: Substitute the definitions into the equation
Substituting the definitions into the equation gives us:
\[ \cot x + \tan x = \frac{\cos x}{\sin x} + \frac{\sin x}{\cos x} \]

### Step 3: Find a common denominator
The common denominator for the two fractions is \( \sin x \cos x \). Thus, we can rewrite the equation as:
\[ \frac{\cos^2 x + \sin^2 x}{\sin x \cos x} \]

### Step 4: Use the Pythagorean identity
We know from the Pythagorean identity that:
\[ \cos^2 x + \sin^2 x = 1 \]
Substituting this into our equation gives:
\[ \frac{1}{\sin x \cos x} \]

### Step 5: Rewrite the expression
We can rewrite \( \frac{1}{\sin x \cos x} \) using the identities for cosecant and secant:
\[ \frac{1}{\sin x \cos x} = \csc x \sec x \]

### Conclusion
Thus, we find that:
\[ \cot x + \tan x = \csc x \sec x \]