`cosec A -2 cot 2A cos A=`

Options

A. `2 sin A `
B. `sec A `
C. `2 cos A cot A`
D. `cos A `

Answer

To solve the expression \( \csc A - 2 \cot 2A \cos A \), we will break it down step by step.

### Step 1: Rewrite the terms in terms of sine and cosine
We know that:
- \( \csc A = \frac{1}{\sin A} \)
- \( \cot 2A = \frac{\cos 2A}{\sin 2A} \)

Thus, we can rewrite the expression:
\[ \csc A - 2 \cot 2A \cos A = \frac{1}{\sin A} - 2 \left( \frac{\cos 2A}{\sin 2A} \right) \cos A \]

### Step 2: Substitute the double angle identities
Using the double angle formulas:
- \( \cos 2A = 2 \cos^2 A - 1 \)
- \( \sin 2A = 2 \sin A \cos A \)

Substituting these into the expression gives:
\[ \frac{1}{\sin A} - 2 \left( \frac{2 \cos^2 A - 1}{2 \sin A \cos A} \right) \cos A \]

### Step 3: Simplify the second term
The second term simplifies as follows:
\[ - 2 \left( \frac{2 \cos^2 A - 1}{2 \sin A \cos A} \right) \cos A = - \frac{(2 \cos^2 A - 1) \cos A}{\sin A} \]

### Step 4: Combine the terms
Now we can combine the two terms:
\[ \frac{1}{\sin A} - \frac{(2 \cos^2 A - 1) \cos A}{\sin A} = \frac{1 - (2 \cos^2 A - 1) \cos A}{\sin A} \]

### Step 5: Simplify the numerator
The numerator simplifies to:
\[ 1 - (2 \cos^2 A - 1) \cos A = 1 - 2 \cos^3 A + \cos A \]
Thus, we have:
\[ \frac{1 + \cos A - 2 \cos^3 A}{\sin A} \]

### Step 6: Final expression
The final expression is:
\[ \frac{1 + \cos A - 2 \cos^3 A}{\sin A} \]