If `o^(@)leAle90^(@)`, the simplified form of the given expression sin A cos A(tan A-cotA) is

Options

A. 1
B. `1-2sin^(2)A`
C. `2sin^(2)A-1`
D. `1-cos^(2)A`

Answer

To simplify the expression \( \sin A \cos A (\tan A - \cot A) \), we can follow these steps:

### Step 1: Rewrite \(\tan A\) and \(\cot A\)
We know that:
\[ \tan A = \frac{\sin A}{\cos A} \quad \text{and} \quad \cot A = \frac{\cos A}{\sin A} \]
So, we can rewrite the expression:
\[ \tan A - \cot A = \frac{\sin A}{\cos A} - \frac{\cos A}{\sin A} \]

### Step 2: Find a common denominator
To combine the two fractions, we need a common denominator:
\[ \tan A - \cot A = \frac{\sin^2 A - \cos^2 A}{\sin A \cos A} \]

### Step 3: Substitute back into the original expression
Now, substitute this back into the original expression:
\[ \sin A \cos A \left( \tan A - \cot A \right) = \sin A \cos A \left( \frac{\sin^2 A - \cos^2 A}{\sin A \cos A} \right) \]

### Step 4: Simplify the expression
The \(\sin A \cos A\) in the numerator and denominator cancels out:
\[ = \sin^2 A - \cos^2 A \]

### Step 5: Use the Pythagorean identity
We can express \(\cos^2 A\) in terms of \(\sin^2 A\):
\[ \cos^2 A = 1 - \sin^2 A \]
Thus:
\[ \sin^2 A - \cos^2 A = \sin^2 A - (1 - \sin^2 A) = \sin^2 A - 1 + \sin^2 A = 2\sin^2 A - 1 \]

### Final Result
The simplified form of the expression \( \sin A \cos A (\tan A - \cot A) \) is:
\[ 2\sin^2 A - 1 \]

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