Choose the correct answer in each of the following questions:
`sec^(2)10^(@)-cot^(2)80^(@)=?`

Options

A. 1
B. 0
C. `(3)/(2)`
D. `(1)/(2)`

Explanation

To solve the expression \( \sec^2(10^\circ) - \cot^2(80^\circ) \), we can follow these steps:

### Step 1: Rewrite \(\cot(80^\circ)\)
We know that:
\[ \cot(80^\circ) = \tan(90^\circ - 80^\circ) = \tan(10^\circ) \]
Thus, we can rewrite \(\cot^2(80^\circ)\) as:
\[ \cot^2(80^\circ) = \tan^2(10^\circ) \]

### Step 2: Substitute into the expression
Now we can substitute this back into the original expression:
\[ \sec^2(10^\circ) - \cot^2(80^\circ) = \sec^2(10^\circ) - \tan^2(10^\circ) \]

### Step 3: Use the Pythagorean identity
We know from trigonometric identities that:
\[ \sec^2(\theta) - \tan^2(\theta) = 1 \]
for any angle \(\theta\). Therefore, we can apply this identity:
\[ \sec^2(10^\circ) - \tan^2(10^\circ) = 1 \]

### Final Answer
Thus, the value of \( \sec^2(10^\circ) - \cot^2(80^\circ) \) is:
\[ \boxed{1} \]

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