Question

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

• A. $1$
• B. $\cos \theta$
• C. $\sin \theta$
• D. $2 \cos \theta$
• E. $0$

Hint:

Shortcut Tip: You can also solve this by simply choosing an arbitrary value for $\theta$, like $\theta = 0^\circ$. The expression becomes $\sin(45^\circ) - \cos(45^\circ) = \frac{1}{\sqrt{2}} - \frac{1}{\sqrt{2}} = 0$.

Answer 1

Answer: (E)

Concept:
The co-function identity states that the sine of an angle is equal to the cosine of its complement: $\sin(x) = \cos(90^\circ - x)$. By treating $(45^\circ + \theta)$ as a single angle, we can convert the sine term directly into a cosine term to compare them.

Step 1: State the given expression.
We need to evaluate the difference: $$\sin(45^\circ + \theta) - \cos(45^\circ - \theta)$$

Step 2: Apply the co-function identity to the sine term.
Using $\sin(A) = \cos(90^\circ - A)$, let $A = 45^\circ + \theta$: $$\sin(45^\circ + \theta) = \cos(90^\circ - (45^\circ + \theta))$$

Step 3: Simplify the angle inside the cosine.
Distribute the negative sign inside the parentheses and combine the degree terms: $$= \cos(90^\circ - 45^\circ - \theta)$$ $$= \cos(45^\circ - \theta)$$ We have proven that $\sin(45^\circ + \theta)$ is exactly equal to $\cos(45^\circ - \theta)$.

Step 4: Substitute back into the original expression.
Replace the first term of our original expression with its new equivalent form: $$\cos(45^\circ - \theta) - \cos(45^\circ - \theta)$$

Step 5: Calculate the final numerical answer.
Since we are subtracting a quantity from an identical quantity, the result is zero: $$= 0$$ Hence the correct answer is (E) $0$.