Assertion (A) : Value of `sec^(2) 10^(@) -cot^(2) 80^(@)` is 1
Reason (R ) : Value of sin `30^(@) = 1/2 `

Options

A. Both A and R are true and R is the correct explanation of A.
B. Both A and R are true and R is NOT the correct explanation of A.
C. A is true but R is false
D. A is false but R is true

Answer

To solve the assertion and reason problem, we will analyze both the assertion (A) and the reason (R) step by step.

### Step 1: Analyze the Assertion (A)
The assertion states that:
\[ \sec^2(10^\circ) - \cot^2(80^\circ) = 1 \]

#### Step 1.1: Use the identity for cotangent
We know that:
\[ \cot(80^\circ) = \tan(90^\circ - 80^\circ) = \tan(10^\circ) \]

#### Step 1.2: Substitute cotangent in the assertion
Thus, we can rewrite the assertion as:
\[ \sec^2(10^\circ) - \tan^2(10^\circ) = 1 \]

#### Step 1.3: Use the Pythagorean identity
We know from trigonometric identities that:
\[ \sec^2(\theta) - \tan^2(\theta) = 1 \]
for any angle \(\theta\).

#### Step 1.4: Apply the identity to our assertion
So, applying this identity:
\[ \sec^2(10^\circ) - \tan^2(10^\circ) = 1 \]
This confirms that the assertion is true.

### Step 2: Analyze the Reason (R)
The reason states that:
\[ \sin(30^\circ) = \frac{1}{2} \]

#### Step 2.1: Verify the value of sin(30°)
From trigonometric tables or basic trigonometric knowledge:
\[ \sin(30^\circ) = \frac{1}{2} \]
This is also true.

### Conclusion
Both the assertion (A) and the reason (R) are true. However, the reason does not directly support the assertion since they are discussing different trigonometric functions.

### Final Answer
- Assertion (A) is true.
- Reason (R) is true.
- However, the reason does not explain the assertion.