Assertion (A): `sec^(2) 23^(@) - tan^(2) 23^(@) =1`
Reason ( R): For every real value of `theta, sec^(2) theta - tan^(2) theta =1`

Options

A. both A and R are true and R is a correct explanation of A.
B. both A and R are true but R is not a correct explanation of A.
C. A is true, R is false
D. A is false, R is true

Answer

To solve the assertion and reason provided in the question, we will analyze both parts step by step.

### Step 1: Analyze the Assertion (A)
The assertion states that:
\[ \sec^2(23^\circ) - \tan^2(23^\circ) = 1 \]

Using the Pythagorean identity in trigonometry, we know that:
\[ \sec^2(\theta) - \tan^2(\theta) = 1 \]
This identity holds true for any angle \(\theta\) where \(\sec(\theta)\) and \(\tan(\theta)\) are defined.

Since \(23^\circ\) is not an odd multiple of \(90^\circ\) (or \(\frac{\pi}{2}\)), both \(\sec(23^\circ)\) and \(\tan(23^\circ)\) are defined. Therefore, we can conclude:
\[ \sec^2(23^\circ) - \tan^2(23^\circ) = 1 \]
Thus, the assertion (A) is **correct**.

### Step 2: Analyze the Reason (R)
The reason states that:
"For every real value of \(\theta\), \(\sec^2(\theta) - \tan^2(\theta) = 1\)".

While the statement is true, we need to consider the conditions under which it holds. The identity \(\sec^2(\theta) - \tan^2(\theta) = 1\) is valid for all \(\theta\) except for those angles where \(\sec(\theta)\) and \(\tan(\theta)\) are undefined. Specifically, these are the angles of the form:
\[ \theta = \frac{(2n + 1)\pi}{2} \]
for any integer \(n\).

Since \(23^\circ\) does not fall into this category, the reason (R) is **incorrect** as it implies that the identity holds for every real value of \(\theta\) without exception.

### Conclusion
- Assertion (A) is **correct**.
- Reason (R) is **incorrect**.

### Final Answer
- Assertion (A) is true.
- Reason (R) is false.

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