How do you use sum to product formulas to write the sum or difference #sin(x+pi/4)-sin(x-pi/4)# as a product?

Question

Asher Friedman · Accepted Answer

#sin(x+pi/4)-sin(x-pi/4)=sqrt2cosx# As #sin(A+B)=sinAcosB+cosAsinB# and #sin(A-B)=sinAcosB-cosAsinB# subtracting second from first we get #sin(A+B)-sin(A-B)=2cosAsinB# Hence #sin(x+pi/4)-sin(x-pi/4)=2cosxsin(pi/4)# = #2cosx xx1/sqrt2=sqrt2cosx#

Jordan Armstrong · Answer

undefined To express $ \sin(x + \frac{\pi}{4}) - \sin(x - \frac{\pi}{4}) $ as a product, we utilize the sum-to-product formula for sine, which states:

$$ \sin(A) - \sin(B) = 2\sin\left(\frac{A-B}{2}\right)\cos\left(\frac{A+B}{2}\right) $$

Applying this formula to the given expression, we have:

$$ \sin(x + \frac{\pi}{4}) - \sin(x - \frac{\pi}{4}) = 2\sin\left(\frac{(x + \frac{\pi}{4}) - (x - \frac{\pi}{4})}{2}\right)\cos\left(\frac{(x + \frac{\pi}{4}) + (x - \frac{\pi}{4})}{2}\right) $$

$$ = 2\sin\left(\frac{\frac{\pi}{2}}{2}\right)\cos\left(\frac{2x}{2}\right) $$

$$ = 2\sin\left(\frac{\pi}{4}\right)\cos(x) $$

So, $ \sin(x + \frac{\pi}{4}) - \sin(x - \frac{\pi}{4}) $ can be expressed as $ 2\sin\left(\frac{\pi}{4}\right)\cos(x) $.