Question

Simplify \( \cos\left(\dfrac{3\pi}{4}+x\right) - \sin\left(\dfrac{\pi}{4}-x\right) \)

• A. \( -\sqrt{2}\cos x \)
• B. \( -\sqrt{2}\sin x \)
• C. \( \sqrt{2}\cos x \)
• D. \( \sqrt{2}\sin x \)

Hint:

Always apply compound angle formulas carefully and simplify step by step to avoid sign errors.

Answer 1

The Correct Option is A

Step 1: Expand the cosine term.
\[ \cos\left(\frac{3\pi}{4}+x\right) = \cos\frac{3\pi}{4}\cos x - \sin\frac{3\pi}{4}\sin x \] \[ = -\frac{1}{\sqrt{2}}\cos x - \frac{1}{\sqrt{2}}\sin x \] Step 2: Expand the sine term.
\[ \sin\left(\frac{\pi}{4}-x\right) = \sin\frac{\pi}{4}\cos x - \cos\frac{\pi}{4}\sin x \] \[ = \frac{1}{\sqrt{2}}\cos x - \frac{1}{\sqrt{2}}\sin x \] Step 3: Subtract the expressions.
\[ \left(-\frac{1}{\sqrt{2}}\cos x - \frac{1}{\sqrt{2}}\sin x\right) - \left(\frac{1}{\sqrt{2}}\cos x - \frac{1}{\sqrt{2}}\sin x\right) \] Step 4: Simplify.
\[ = -\frac{2}{\sqrt{2}}\cos x = -\sqrt{2}\cos x \] Step 5: Conclusion.
Hence, the simplified expression is \( -\sqrt{2}\cos x \).