Statement-1: `cos36^(@)gttan36^(@)`
Statement-2: `cos36^(@)gtsin36^(@)`

Options

A. A
B. B
C. C
D. D

Answer

To solve the problem, we need to analyze the two statements given regarding the trigonometric functions at an angle of 36 degrees.

### Step-by-step Solution:

**Step 1: Analyze Statement 1**
- We need to evaluate whether \( \cos(36^\circ) > \tan(36^\circ) \).
- Recall the definitions of the trigonometric functions:
- \( \tan(36^\circ) = \frac{\sin(36^\circ)}{\cos(36^\circ)} \).
- Therefore, the inequality can be rewritten as:
\[ \cos(36^\circ) > \frac{\sin(36^\circ)}{\cos(36^\circ)} \]
- Multiplying both sides by \( \cos(36^\circ) \) (which is positive for \( 36^\circ \)):
\[ \cos^2(36^\circ) > \sin(36^\circ) \]

**Step 2: Use the Pythagorean Identity**
- We know from the Pythagorean identity that:
\[ \sin^2(36^\circ) + \cos^2(36^\circ) = 1 \]
- Therefore, we can express \( \sin(36^\circ) \) in terms of \( \cos(36^\circ) \):
\[ \sin(36^\circ) = \sqrt{1 - \cos^2(36^\circ)} \]

**Step 3: Compare Values**
- We need to determine if:
\[ \cos^2(36^\circ) > \sqrt{1 - \cos^2(36^\circ)} \]
- To analyze this, we can use numerical values or a calculator to find \( \cos(36^\circ) \) and \( \tan(36^\circ) \):
- \( \cos(36^\circ) \approx 0.809 \)
- \( \tan(36^\circ) \approx 0.726 \)
- Since \( 0.809 > 0.726 \), we conclude that:
\[ \cos(36^\circ) > \tan(36^\circ) \]
- Thus, Statement 1 is **True**.

**Step 4: Analyze Statement 2**
- Now we need to evaluate whether \( \cos(36^\circ) > \sin(36^\circ) \).
- We can use the same numerical values:
- \( \sin(36^\circ) \approx 0.588 \)
- Since \( 0.809 > 0.588 \), we conclude that:
\[ \cos(36^\circ) > \sin(36^\circ) \]
- Thus, Statement 2 is also **True**.

### Conclusion
Both statements are true:
- Statement 1: \( \cos(36^\circ) > \tan(36^\circ) \) is true.
- Statement 2: \( \cos(36^\circ) > \sin(36^\circ) \) is true.