What is the value of sec `(90^@-A)-cot A cos (90^@-A)tan(90^@-A)` ?

Answer

To solve the expression \( \sec(90^\circ - A) - \cot A \cdot \cos(90^\circ - A) \cdot \tan(90^\circ - A) \), we will use trigonometric identities.

### Step 1: Rewrite the trigonometric functions
Using the co-function identities:
- \( \sec(90^\circ - A) = \csc A \)
- \( \cos(90^\circ - A) = \sin A \)
- \( \tan(90^\circ - A) = \cot A \)

Substituting these into the expression gives:
\[ \sec(90^\circ - A) - \cot A \cdot \cos(90^\circ - A) \cdot \tan(90^\circ - A) = \csc A - \cot A \cdot \sin A \cdot \cot A \]

### Step 2: Simplify the expression
Now, simplify the term \( \cot A \cdot \sin A \cdot \cot A \):
\[ \cot A = \frac{\cos A}{\sin A} \]
Thus,
\[ \cot A \cdot \sin A \cdot \cot A = \frac{\cos A}{\sin A} \cdot \sin A \cdot \frac{\cos A}{\sin A} = \frac{\cos^2 A}{\sin A} \]

Now, substituting this back into the expression:
\[ \csc A - \frac{\cos^2 A}{\sin A} \]

### Step 3: Rewrite \(\csc A\)
Recall that \( \csc A = \frac{1}{\sin A} \). Therefore, we can rewrite the expression:
\[ \frac{1}{\sin A} - \frac{\cos^2 A}{\sin A} \]

### Step 4: Combine the fractions
Since both terms have the same denominator, we can combine them:
\[ \frac{1 - \cos^2 A}{\sin A} \]

### Step 5: Use the Pythagorean identity
Using the identity \( \sin^2 A + \cos^2 A = 1 \), we know that:
\[ 1 - \cos^2 A = \sin^2 A \]

Thus, we can substitute:
\[ \frac{\sin^2 A}{\sin A} = \sin A \]

### Final Answer
The value of the expression is:
\[ \sin A \]