Trigonometry/Sum into Product

These are exercises on the formulae derived in Book 1 for converting the sum or difference of two sines or two cosines into a product.

[1] $\frac{\sin 7\theta -\sin 5\theta }{\cos 7\theta +\cos 5\theta }=\tan \theta$

---

[2]

$\frac{\cos 6\alpha -\cos 4\alpha }{\sin 6\alpha +\sin 4\alpha }=-\tan \alpha$

---

[3]

$\frac{\sin A+\sin 3A}{\cos A+\cos 3A}=\tan 2A$

---

[4]

$\frac{\sin 7A-\sin A}{\sin 8A-\sin 2A}=\cos 4A\sec 5A$

---

[5]

$\frac{\cos 2\varphi +\cos 2\theta }{\cos 2\varphi -\cos 2\theta }=\cot (\varphi +\theta )\cot (\varphi -\theta )$

---

[6]

$\frac{\sin 2A+\sin 2B}{\sin 2A-\sin 2B}=\tan (A-B)\cot (A-B)$

---

[7]

$\frac{\sin A+\sin 2A}{\cos A-\cos 2A}=\cot \left(\frac{A}{2}\right)$

---

[8]

$\frac{\sin 5\lambda -\sin 3\lambda }{\cos 5\lambda +\cos 3\lambda }=\tan \lambda$

---

[9]

$\frac{\cos 2B-\cos 2A}{\sin 2B+\sin 2A}=\tan (A-B)$

---

[10]

$\cos (\varphi +\theta )+\sin (\varphi -\theta )=2\sin (45^o+\varphi )\cos (45^o+\theta )$

---

[11]

$\frac{\sin \alpha +\sin \beta }{\sin \alpha -\sin \beta }=\tan \left(\frac{\alpha +\beta }{2}\right)\cot \left(\frac{\alpha -\beta }{2}\right)$

---

[12]

$\frac{\cos \psi +\cos \omega }{\cos \omega -\cos \psi }=\cot \left(\frac{\psi +\omega }{2}\right)\cot \left(\frac{\psi -\omega }{2}\right)$

---

[13]

$\frac{\sin \varphi +\sin \theta }{\cos \varphi +\cos \theta }=\tan \left(\frac{\varphi +\theta }{2}\right)$

---

[14]

$\frac{\sin A-\sin B}{\cos B-\cos A}=\cot \left(\frac{A+B}{2}\right)$

---

[15]

$\frac{\cos 3A-\cos A}{\sin 3A-\sin A}+\frac{\cos 2A-\cos 4A}{\sin 4A-\sin 2A}=\frac{\sin A}{\cos 2A\cos 3A}$

---

[16]

$a\cos \varphi +b\sin \varphi =\sqrt{a^2+b^2}\cos [\varphi -{\tan }^{-1}(\frac{b}{a})]$