Evaluate `((-101)/(cos^2A)+(101)/(cot^2A))`

Options

A. 101
B. `-101`
C. 1
D. `-1`

Answer

To evaluate the expression \((-101/\cos^2 A + 101/\cot^2 A)\), we will follow these steps:

### Step 1: Rewrite \(\cot^2 A\)
We know that \(\cot A = \frac{\cos A}{\sin A}\). Therefore, \(\cot^2 A = \frac{\cos^2 A}{\sin^2 A}\).

### Step 2: Substitute \(\cot^2 A\) in the expression
The expression now becomes:
\[ -\frac{101}{\cos^2 A} + \frac{101}{\frac{\cos^2 A}{\sin^2 A}} \]
This simplifies to:
\[ -\frac{101}{\cos^2 A} + \frac{101 \sin^2 A}{\cos^2 A} \]

### Step 3: Combine the terms
Now, we can combine the two fractions:
\[ -\frac{101}{\cos^2 A} + \frac{101 \sin^2 A}{\cos^2 A} = \frac{-101 + 101 \sin^2 A}{\cos^2 A} \]
This can be factored as:
\[ \frac{101 (\sin^2 A - 1)}{\cos^2 A} \]

### Step 4: Use the Pythagorean identity
We know that \(\sin^2 A + \cos^2 A = 1\), which means \(\sin^2 A - 1 = -\cos^2 A\). Substituting this in gives:
\[ \frac{101 (-\cos^2 A)}{\cos^2 A} \]

### Step 5: Simplify the expression
The \(\cos^2 A\) in the numerator and denominator cancels out:
\[ -101 \]

### Final Answer
Thus, the value of the expression \((-101/\cos^2 A + 101/\cot^2 A)\) is:
\[ \boxed{-101} \]