Question

Please help me with this question I don’t understand and been struggling would love to appreciate your help this is not test or anything it’s just Hw thank you!

Answer 1

we have the expression

$\sin (x-\frac{\pi}{4})=\frac{\sqrt[]{2}}{2}(\sin x-\cos x)$

Remember that

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

and

$\cos \frac{\pi}{4}=\sin \frac{\pi}{4}=\frac{\sqrt[]{2}}{2}$

substitute

$\sin (x-\frac{\pi}{4})=\sin x\cdot\frac{\sqrt[]{2}}{2}-\cos x\cdot\frac{\sqrt[]{2}}{2}$

Factor

$\sin (x-\frac{\pi}{4})=\frac{\sqrt[]{2}}{2}\cdot(\sin x-\cos x)$

therefore

The identity is verified

the answer in the blank box must be

$\sin x\cdot\cos \frac{\pi}{4}-\cos x\cdot\sin \frac{\pi}{4}$