If `0^(@)lethetale90^(@)` then for any value of `theta` which one is correct ?

Options

A. `sintheta=sqrt(2)`
B. `sintheta+costheta=2`
C. `sintheta+costheta=0`
D. `sintheta-costheta=1`

Answer

To solve the problem, we need to analyze the given options based on the properties of sine and cosine functions for angles between 0° and 90°.

### Step-by-Step Solution:

1. **Understanding the Range of Sine and Cosine:**
- For any angle \( \theta \) where \( 0^\circ \leq \theta \leq 90^\circ \), both \( \sin \theta \) and \( \cos \theta \) will have values between 0 and 1.
- Therefore, \( \sin \theta \) can take values from 0 to 1, and \( \cos \theta \) can also take values from 0 to 1.

**Hint:** Remember that sine and cosine functions are non-negative in the first quadrant.

2. **Analyzing Option A: \( \sin \theta = \sqrt{2} \)**
- The value \( \sqrt{2} \) is approximately 1.414, which is greater than 1.
- Since \( \sin \theta \) cannot exceed 1, this option is incorrect.

**Hint:** Check if the value is within the range of sine function.

3. **Analyzing Option B: \( \sin \theta + \cos \theta = 2 \)**
- The maximum value of \( \sin \theta + \cos \theta \) occurs when both sine and cosine are at their maximum, which is 1.
- Thus, \( \sin \theta + \cos \theta \) can be at most \( 1 + 1 = 2 \), but this occurs only when both sine and cosine are 1, which is not possible for any angle \( \theta \) in the given range.
- Therefore, this option is also incorrect.

**Hint:** Consider the maximum possible values of sine and cosine.

4. **Analyzing Option C: \( \sin \theta + \cos \theta = 0 \)**
- The only way for the sum of sine and cosine to equal zero is if both are zero.
- However, \( \sin \theta = 0 \) only at \( \theta = 0^\circ \) and \( \cos \theta = 0 \) only at \( \theta = 90^\circ \).
- These values cannot occur simultaneously, so this option is incorrect.

**Hint:** Think about when sine and cosine can both be zero.

5. **Analyzing Option D: \( \sin \theta - \cos \theta = 1 \)**
- Rearranging gives \( \sin \theta = \cos \theta + 1 \).
- At \( \theta = 90^\circ \), \( \sin 90^\circ = 1 \) and \( \cos 90^\circ = 0 \), so \( 1 - 0 = 1 \).
- This condition holds true, making this option valid.

**Hint:** Check specific angles to see if the equation holds true.

### Conclusion:
The correct option is **D: \( \sin \theta - \cos \theta = 1 \)**, which is valid when \( \theta = 90^\circ \).