Question

Prove this identity! IMAGE ATTACHED!

Answer 1

\(\Large\tt \begin{align} \color{black}{sin\theta-1\\~\\
=-(1-sin\theta)\\~\\
=-(1-2sin\frac{\theta}{2}cos\frac{\theta}{2})\\~\\
=-(cos^2\frac{\theta}{2}+sin^2\frac{\theta}{2}-2sin\frac{\theta}{2}cos\frac{\theta}{2})\\~\\
=-(cos\frac{\theta}{2}-sin\frac{\theta}{2})^2\\~\\
=-2(\frac{1}{\sqrt{2}}cos\frac{\theta}{2}-\frac{1}{\sqrt{2}}sin\frac{\theta}{2})^2\\~\\
=-2(sin45^{\circ}cos\frac{\theta}{2}-cos45^{\circ}sin\frac{\theta}{2})^2\\~\\
=-2(sin45^{\circ}-\frac{\theta}{2})^2\\~\\
=-2sin^2(45^{\circ}-\frac{\theta}{2})\\~\\

}\end{align}\)