If sin`x^@ = a`, which of the following must be true for all values of x ?

Options

A. `cos x^@=a`
B. `sin(90^@ -x^@)=a`
C. `cos (90^@ -x^@)=a`
D. `sin(x^2)^@=a^2`

Explanation

To solve the problem, we need to analyze the given equation \( \sin x = a \) and determine which statements must be true for all values of \( x \).

### Step-by-Step Solution:

1. **Understanding the Given Equation**:
We start with the equation \( \sin x = a \). This means that the sine of angle \( x \) is equal to some value \( a \).

**Hint**: Remember that the sine function has a range of values between -1 and 1, so \( a \) must satisfy \( -1 \leq a \leq 1 \).

2. **Using the Complementary Angle Identity**:
We know from trigonometric identities that:
\[ \cos(90^\circ - x) = \sin x \]
This means that the cosine of the complementary angle (90 degrees minus \( x \)) is equal to the sine of \( x \).

**Hint**: Recall that the sine and cosine functions are related through complementary angles.

3. **Substituting the Given Value**:
Since we have \( \sin x = a \), we can substitute \( a \) into the complementary angle identity:
\[ \cos(90^\circ - x) = a \]

**Hint**: This substitution shows how sine and cosine are connected through the complementary angle.

4. **Conclusion**:
From the above steps, we conclude that:
\[ \cos(90^\circ - x) = a \]
This relationship must hold true for all values of \( x \) where \( \sin x = a \).

**Hint**: Check if this relationship is consistent with any options provided in the question.

### Final Answer:
The statement that must be true for all values of \( x \) is:
\[ \cos(90^\circ - x) = a \]