Question

Find the value of sin (45° +Ø) - cos (45° - Ø).​

Answer 1

Answer:

I hope this will help you.

Answer 2

Answer:

0

Step-by-step explanation:

$\sin \left(45^{\circ \:}+\o\right)-\cos \left(45^{\circ \:}-\o\right)$

$\sin \left(45^{\circ \:}+\o\right)$

$\mathrm{Use\:the\:following\:identity}:\quad \sin \left(s+t\right)=\cos \left(s\right)\sin \left(t\right)+\cos \left(t\right)\sin \left(s\right)$

$=\cos \left(45^{\circ \:}\right)\sin \left(\o\right)+\cos \left(\o\right)\sin \left(45^{\circ \:}\right)$

$\mathrm{Use\:the\:following\:trivial\:identity}:\quad \cos \left(45^{\circ \:}\right)=\dfrac{\sqrt{2}}{2}$

$=\dfrac{\sqrt{2}}{2}\sin \left(\o\right)+\sin \left(45^{\circ \:}\right)\cos \left(\o\right)$

$\mathrm{Use\:the\:following\:trivial\:identity}:\quad \sin \left(45^{\circ \:}\right)=\dfrac{\sqrt{2}}{2}$

$=\dfrac{\sqrt{2}}{2}\sin \left(\o\right)+\dfrac{\sqrt{2}}{2}\cos \left(\o\right)$

$=\dfrac{\sqrt{2}}{2}\sin \left(\o\right)+\dfrac{\sqrt{2}}{2}\cos \left(\o\right)-\cos \left(45^{\circ \:}-\o\right)$

$\cos \left(45^{\circ \:}-\o\right)$

$\mathrm{Use\:the\:following\:identity}:\quad \cos \left(s-t\right)=\cos \left(s\right)\cos \left(t\right)+\sin \left(s\right)\sin \left(t\right)$

$=\cos \left(45^{\circ \:}\right)\cos \left(\o\right)+\sin \left(45^{\circ \:}\right)\sin \left(\o\right)$

$\mathrm{Use\:the\:following\:trivial\:identity}:\quad \cos \left(45^{\circ \:}\right)=\dfrac{\sqrt{2}}{2}$

$=\dfrac{\sqrt{2}}{2}\cos \left(\o\right)+\sin \left(45^{\circ \:}\right)\sin \left(\o\right)$

$\mathrm{Use\:the\:following\:trivial\:identity}:\quad \sin \left(45^{\circ \:}\right)=\dfrac{\sqrt{2}}{2}$

$=\dfrac{\sqrt{2}}{2}\cos \left(\o\right)+\dfrac{\sqrt{2}}{2}\sin \left(\o\right)$

$=\dfrac{\sqrt{2}}{2}\sin \left(\o\right)+\dfrac{\sqrt{2}}{2}\cos \left(\o\right)-\left(\dfrac{\sqrt{2}}{2}\cos \left(\o\right)+\dfrac{\sqrt{2}}{2}\sin \left(\o\right)\right)$

$-\left(\dfrac{\sqrt{2}}{2}\cos \left(\o\right)+\dfrac{\sqrt{2}}{2}\sin \left(\o\right)\right)$

$\mathrm{Distribute\:parentheses}$

$=-\left(\dfrac{\sqrt{2}}{2}\cos \left(\o\right)\right)-\left(\dfrac{\sqrt{2}}{2}\sin \left(\o\right)\right)$

$\mathrm{Apply\:minus-plus\:rules}$

$+\left(-a\right)=-a$

$=-\dfrac{\sqrt{2}}{2}\cos \left(\o\right)-\dfrac{\sqrt{2}}{2}\sin \left(\o\right)$

$=\dfrac{\sqrt{2}}{2}\sin \left(\o\right)+\dfrac{\sqrt{2}}{2}\cos \left(\o\right)-\dfrac{\sqrt{2}}{2}\cos \left(\o\right)-\dfrac{\sqrt{2}}{2}\sin \left(\o\right)$

$\mathrm{Simplify}\:\dfrac{\sqrt{2}}{2}\sin \left(\o\right)+\dfrac{\sqrt{2}}{2}\cos \left(\o\right)-\dfrac{\sqrt{2}}{2}\cos \left(\o\right)-\dfrac{\sqrt{2}}{2}\sin \left(\o\right)$

$\dfrac{\sqrt{2}}{2}\sin \left(\o\right)+\dfrac{\sqrt{2}}{2}\cos \left(\o\right)-\dfrac{\sqrt{2}}{2}\cos \left(\o\right)-\dfrac{\sqrt{2}}{2}\sin \left(\o\right)$

$\mathrm{Add\:similar\:elements:}\:\dfrac{\sqrt{2}}{2}\cos \left(\o\right)-\dfrac{\sqrt{2}}{2}\cos \left(\o\right)=0$

$=\dfrac{\sqrt{2}}{2}\sin \left(\o\right)-\dfrac{\sqrt{2}}{2}\sin \left(\o\right)$

$\mathrm{Add\:similar\:elements:}\:\dfrac{\sqrt{2}}{2}\sin \left(\o\right)-\dfrac{\sqrt{2}}{2}\sin \left(\o\right)=0$

$=0$