Consider the following
I. `sin^2 1^@+ cos^2 1^@ =1`
II. `sec^2 33^@ -cot^2 57^@ = cosec^2 37^@ - tan^2 53^@`
Which of the above statement is/are correct ?

Options

A. Only I
B. Only II
C. Both I and II
D. Neither I nor II

Answer

To determine which of the given statements is correct, we will analyze both statements step by step.

### Statement I:
**sin²(1°) + cos²(1°) = 1**

**Step 1:** Recall the Pythagorean identity in trigonometry, which states that for any angle θ:
\[ \sin^2 \theta + \cos^2 \theta = 1 \]

**Step 2:** Substitute θ with 1°:
\[ \sin^2(1°) + \cos^2(1°) = 1 \]

**Conclusion for Statement I:** This statement is true as it directly follows from the Pythagorean identity.

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### Statement II:
**sec²(33°) - cot²(57°) = cosec²(37°) - tan²(53°)**

**Step 1:** Analyze the left-hand side (LHS):
\[ \text{LHS} = \sec^2(33°) - \cot^2(57°) \]

**Step 2:** Use the identity:
\[ \sec^2 \theta - \tan^2 \theta = 1 \]
Here, we can express cot(57°) in terms of tan(33°):
\[ \cot(57°) = \tan(90° - 57°) = \tan(33°) \]
Thus,
\[ \cot^2(57°) = \tan^2(33°) \]

**Step 3:** Substitute this into the LHS:
\[ \text{LHS} = \sec^2(33°) - \tan^2(33°) = 1 \]

**Step 4:** Now analyze the right-hand side (RHS):
\[ \text{RHS} = \csc^2(37°) - \tan^2(53°) \]

**Step 5:** Use the identity:
\[ \csc^2 \theta - \cot^2 \theta = 1 \]
Here, we can express tan(53°) in terms of cot(37°):
\[ \tan(53°) = \cot(37°) \]
Thus,
\[ \tan^2(53°) = \cot^2(37°) \]

**Step 6:** Substitute this into the RHS:
\[ \text{RHS} = \csc^2(37°) - \cot^2(37°) = 1 \]

**Conclusion for Statement II:** Since both LHS and RHS equal 1, this statement is also true.

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### Final Conclusion:
Both statements I and II are correct.

### Summary of Answers:
- **Statement I:** Correct
- **Statement II:** Correct

Thus, the answer is that both statements are correct.

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