Question

$\sin(\frac{3\pi}{4}+x)+\cos(\frac{3\pi}{4}+x)-\sin(\frac{3\pi}{4}-x)-\cos(\frac{3\pi}{4}-x)=$

• A. $-2\sqrt{2}\sin x$
• B. $\sqrt{2}\sin x$
• C. $2\sqrt{2}\sin x$
• D. $3\sqrt{2}\sin x$
• E. $-\sqrt{2}\sin x$

Hint:

Math Tip: Grouping terms strategically makes complex trig problems simpler. Instead of fully expanding $\sin(A+B)$ into four terms, using the transformation formulas ($\sin C - \sin D$) condenses the expression immediately.

Answer 1

The Correct Option is A

Concept:
Trigonometry - Sum and Difference Formulas.
Step 1: Group similar trigonometric functions together.
Rearrange the terms in the given expression to group the sine terms and the cosine terms: $$ \left[ \sin\left(\frac{3\pi}{4}+x\right) - \sin\left(\frac{3\pi}{4}-x\right) \right] + \left[ \cos\left(\frac{3\pi}{4}+x\right) - \cos\left(\frac{3\pi}{4}-x\right) \right] $$
Step 2: Apply trigonometric transformation identities.
Recall the standard formulas for the difference of sines and cosines:

• $\sin(A+B) - \sin(A-B) = 2\cos A \sin B$
• $\cos(A+B) - \cos(A-B) = -2\sin A \sin B$

Step 3: Simplify the sine group.
Apply the first identity where $A = \frac{3\pi}{4}$ and $B = x$: $$ \sin\left(\frac{3\pi}{4}+x\right) - \sin\left(\frac{3\pi}{4}-x\right) = 2\cos\left(\frac{3\pi}{4}\right)\sin(x) $$
Step 4: Simplify the cosine group.
Apply the second identity where $A = \frac{3\pi}{4}$ and $B = x$: $$ \cos\left(\frac{3\pi}{4}+x\right) - \cos\left(\frac{3\pi}{4}-x\right) = -2\sin\left(\frac{3\pi}{4}\right)\sin(x) $$
Step 5: Substitute standard trigonometric values.
Evaluate the trigonometric functions at $\frac{3\pi}{4}$ (which is $135^{\circ}$ in the second quadrant):

• $\cos\left(\frac{3\pi}{4}\right) = -\frac{1}{\sqrt{2}}$
• $\sin\left(\frac{3\pi}{4}\right) = \frac{1}{\sqrt{2}}$

Step 6: Calculate the final expression.
Substitute these values back into our simplified groups and add them together: $$ 2\left(-\frac{1}{\sqrt{2}}\right)\sin x + \left(-2\right)\left(\frac{1}{\sqrt{2}}\right)\sin x $$ $$ = -\sqrt{2}\sin x - \sqrt{2}\sin x $$ $$ = -2\sqrt{2}\sin x $$