Simplifying $\sin(\frac{7A}{2}+15^{\circ})\sin(\frac{3A}{2}-75^{\circ})+\cos(\frac{7A}{2}+15^{\circ})\cos(\frac{3A}{2}-75^{\circ})$

Question

Can someone help me with simplifying this expression:

$$ \sin \left( \dfrac{7A}{2} + 15^{\circ} \right) \sin \left( \dfrac{3A}{2} - 75^{\circ} \right) + \cos \left( \dfrac{7A}{2} + 15^{\circ} \right) \cos \left( \dfrac{3A}{2} - 75^{\circ} \right) $$

I know that $\sin x \sin y + \cos x \cos y = \cos(x-y)$, but this way I only get the expression:

$$\cos(2A+100^{\circ})$$

The problem is that I have to find the correct solution among the following ones:

• $-2\sin A \cos A$

• $\cos ^2 A - \sin ^2 A$

• $- \sin A$

• $\cos A$

Maybe I shouldn't have used this trigonometric identity but another one?

Answer 1

Your strategy is correct, but you made an arithmetic mistake.

If we set $$x = \frac{3A}{2} - 75^\circ$$ and $$y = \frac{7A}{2} + 15^\circ$$ then $$x - y = -2A - 90^\circ$$ so \begin{align*} \cos(x - y) & = \cos(-2A - 90^\circ)\\ & = \cos(2A + 90^\circ) \end{align*} Can you take it from here?