Question

The value of $\sin(45^{\circ} + \theta) - \cos(45^{\circ} - \theta)$ is:

• A. 0
• B. $2\sin\theta$
• C. $\sqrt{2}\cos\theta$
• D. 1

Hint:

$45^{\circ} + \theta$ and $45^{\circ} - \theta$ are complementary angles as their sum is $90^{\circ}$.

Answer 1

The Correct Option is A

Step 1: Understand the Mathematical Concept
This problem utilizes the trigonometric identity for complementary angles. Specifically, it uses the rule that the sine of an angle is equal to the cosine of its complement. This is expressed by the formula:
$$\sin(A) = \cos(90^{\circ} - A)$$ By using this identity, we can convert one trigonometric ratio into another to simplify the expression.

Step 2: Analysis & Transformation
We are given the expression: $\sin(45^{\circ} + \theta) - \cos(45^{\circ} - \theta)$.
Let’s transform the first term, $\sin(45^{\circ} + \theta)$, using our identity where $A = (45^{\circ} + \theta)$:
$$\sin(45^{\circ} + \theta) = \cos(90^{\circ} - (45^{\circ} + \theta))$$ Distributing the negative sign inside the parentheses:
$$\cos(90^{\circ} - 45^{\circ} - \theta) = \cos(45^{\circ} - \theta)$$ Now we can see that the first term in our original expression is mathematically identical to the second term.

Step 3: Conclusion
Now, substitute our transformed value back into the original expression:
The expression becomes: $\cos(45^{\circ} - \theta) - \cos(45^{\circ} - \theta)$.
Since we are subtracting a value from itself, the result is 0.

Final Answer: (A)