Assertion (A) : `sin^(2) 67^(@) +cos^(2) 67^(@)=1`
Reason (R ) : For any value of `theta , sin^(2) theta +cos^(2) theta = 1 `

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 given assertion and reason, we will follow these steps:

### Step 1: Understand the Assertion
The assertion states that \( \sin^2 67^\circ + \cos^2 67^\circ = 1 \).

### Step 2: Recall the Trigonometric Identity
We know from trigonometric identities that for any angle \( \theta \):
\[ \sin^2 \theta + \cos^2 \theta = 1 \]

### Step 3: Apply the Identity to the Assertion
In our case, we can let \( \theta = 67^\circ \). Therefore, substituting \( \theta \) into the identity gives us:
\[ \sin^2 67^\circ + \cos^2 67^\circ = 1 \]
This confirms that the assertion is true.

### Step 4: Understand the Reason
The reason states that for any value of \( \theta \), \( \sin^2 \theta + \cos^2 \theta = 1 \). This is indeed a fundamental identity in trigonometry and holds true for all angles \( \theta \).

### Step 5: Conclusion
Since both the assertion and the reason are true, and the reason correctly explains the assertion, we can conclude that both the assertion (A) and the reason (R) are true, and R is the correct explanation for A.

### Final Answer:
Both Assertion (A) and Reason (R) are true, and R is the correct explanation for A.

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