`cosA(secA-cosA)(cotA+tanA)=?`

Options

A. `tanA`
B. `SecA`
C. `sinA`
D. `cotA`

Answer

To solve the expression \( \cos A (\sec A - \cos A)(\cot A + \tan A) \), we will break it down step by step.

### Step 1: Substitute the Trigonometric Identities
The first step is to substitute the trigonometric identities for secant, cotangent, and tangent:
- \( \sec A = \frac{1}{\cos A} \)
- \( \cot A = \frac{\cos A}{\sin A} \)
- \( \tan A = \frac{\sin A}{\cos A} \)

So, we rewrite the expression:
\[ \cos A \left(\frac{1}{\cos A} - \cos A\right) \left(\frac{\cos A}{\sin A} + \frac{\sin A}{\cos A}\right) \]

### Step 2: Simplify the First Parenthesis
Now, simplify the first parenthesis:
\[ \frac{1}{\cos A} - \cos A = \frac{1 - \cos^2 A}{\cos A} = \frac{\sin^2 A}{\cos A} \]
(using the identity \( \sin^2 A + \cos^2 A = 1 \))

### Step 3: Simplify the Second Parenthesis
Next, simplify the second parenthesis:
\[ \frac{\cos A}{\sin A} + \frac{\sin A}{\cos A} = \frac{\cos^2 A + \sin^2 A}{\sin A \cos A} = \frac{1}{\sin A \cos A} \]
(using the identity \( \cos^2 A + \sin^2 A = 1 \))

### Step 4: Combine the Results
Now, substitute back into the expression:
\[ \cos A \cdot \frac{\sin^2 A}{\cos A} \cdot \frac{1}{\sin A \cos A} \]

### Step 5: Simplify the Expression
This simplifies to:
\[ \frac{\sin^2 A}{\sin A \cos A} = \frac{\sin A}{\cos A} = \tan A \]

### Final Result
Thus, the final result is:
\[ \cos A (\sec A - \cos A)(\cot A + \tan A) = \tan A \]