Aus Online Mathematik Brückenkurs 1

The addition formula for sine gives us that

\displaystyle \sin \left( v+\frac{\pi }{4} \right)=\sin v\centerdot \cos \frac{\pi }{4}+\cos v\centerdot \sin \frac{\pi }{4}.

Because we know from exercise b that \displaystyle \sin v=\sqrt{1-b^{2}} we use that \displaystyle \cos \frac{\pi }{4}=\sin \frac{\pi }{4}=\frac{1}{\sqrt{2}} to obtain

\displaystyle \sin \left( v+\frac{\pi }{4} \right)=\sqrt{1-b^{2}}\centerdot \frac{1}{\sqrt{2}}+b\centerdot \frac{1}{\sqrt{2}}.