1Which expression is equal to \( 2\sin A\cos B \): \( \sin(A+B) - \sin(A-B) \), \( \sin(A+B) + \sin(A-B) \), \( \cos(A+B) + \cos(A-B) \), or \( \cos(A+B) - \sin(A-B) \)?Show Answer

2Which expression is equal to \( \sin A - \sin B \): \( 2\sin\frac{A+B}{2}\cos\frac{A-B}{2} \), \( -2\sin\frac{A+B}{2}\sin\frac{A-B}{2} \), \( 2\cos\frac{A+B}{2}\sin\frac{A-B}{2} \), or \( 2\cos\frac{A+B}{2}\cos\frac{A-B}{2} \)?Show Answer

3Which expression is equal to \( \cos C - \cos D \): \( 2\sin\frac{C+D}{2}\cos\frac{C-D}{2} \), \( 2\cos\frac{C+D}{2}\cos\frac{C-D}{2} \), \( -2\sin\frac{C+D}{2}\sin\frac{D-C}{2} \), or \( -2\sin\frac{C+D}{2}\sin\frac{C-D}{2} \)?Show Answer

4What is \( \cos 6A + \cos 4A \) as a product: \( 2\sin 6A\cos 4A \), \( 2\sin 5A\cos A \), \( 2\cos 5A\sin A \), or \( 2\cos 5A\cos A \)?Show Answer

5Which is equal to \( \sin 60^\circ + \sin 40^\circ \): \( 2\sin 50^\circ\cos 10^\circ \), \( 2\sin 50^\circ\sin 10^\circ \), \( 2\cos 50^\circ\sin 10^\circ \), or \( 2\cos 50^\circ\cos 5^\circ \)?Show Answer

6Express \( \sin 50^\circ \cos 30^\circ \) as a sum.Show Answer

7Express \( \cos 25^\circ \cos 15^\circ \) as a sum.Show Answer

8Express \( \cos 10A \sin 6A \) as a difference.Show Answer

9Express \( \sin 27^\circ \sin 32^\circ \) as a difference.Show Answer

10Express \( \sin 50^\circ + \sin 30^\circ \) as a product.Show Answer

11Express \( \cos 25^\circ + \cos 15^\circ \) as a product.Show Answer

12Express \( \cos 7A - \cos 5A \) as a product.Show Answer

13Express \( \sin 67^\circ - \sin 43^\circ \) as a product.Show Answer

14Prove that \( \cos 50^\circ + \cos 40^\circ = \sqrt{2}\cos 5^\circ \).Show Answer

15Prove that \( \sin 50^\circ - \sin 70^\circ = -\sin 10^\circ \).Show Answer

16Prove that \( \sin 10^\circ + \sin 70^\circ = \sqrt{3}\sin 40^\circ \).Show Answer

17Prove that \( \cos 40^\circ - \sin 40^\circ = \sqrt{2}\sin 5^\circ \).Show Answer

18Prove that \( \sin 37^\circ - \cos 67^\circ = \sqrt{3}\sin 7^\circ \).Show Answer

19Prove that \( \frac{\cos 10^\circ - \sin 10^\circ}{\cos 10^\circ + \sin 10^\circ} = \tan 35^\circ \).Show Answer

20Prove that \( \frac{\cos 9^\circ - \sin 9^\circ}{\cos 9^\circ + \sin 9^\circ} = \cot 54^\circ \).Show Answer

21Prove that \( \frac{\cos 12^\circ - \sin 12^\circ}{\cos 12^\circ + \sin 12^\circ} = \tan 33^\circ \).Show Answer

22Prove that \( \frac{\sin A - \sin B}{\sin A + \sin B} = \cot\frac{A+B}{2}\tan\frac{A-B}{2} \).Show Answer

23Prove that \( \frac{\cos B - \cos A}{\sin A + \sin B} = \tan\frac{A-B}{2} \).Show Answer

24Prove that \( \frac{\sin(60^\circ + A) + \sin(60^\circ - A)}{\cos(60^\circ + A) + \cos(60^\circ - A)} = \sqrt{3} \).Show Answer

25Prove that \( 2\sin(45^\circ + \phi)\cos(45^\circ - \phi) = 1 + \sin 2\phi \).Show Answer

26Prove that \( 2\cos(45^\circ + \theta)\cos(45^\circ - \theta) = \cos 2\theta \).Show Answer

27Prove that \( \frac{\sin\theta + \sin 3\theta - \sin 2\theta}{\cos\theta - \cos 2\theta + \cos 3\theta} = \tan 2\theta \).Show Answer

28Prove that \( \frac{\sin 2A + \sin 3A + \sin 5A + \sin 6A}{\cos 2A + \cos 3A + \cos 5A + \cos 6A} = \tan 4A \).Show Answer

29Prove that \( \frac{\sin A - \sin 3A + \sin 5A - \sin 7A}{\cos A - \cos 3A - \cos 5A + \cos 7A} = \cot 2A \).Show Answer

30Prove that \( \frac{\cos 3A\cos 2A - \sin 5A\sin 7A}{\sin 4A\sin 3A - \sin 2A\sin 5A} = 4\sin 5A\cos A \).Show Answer

31Prove that \( 4\cos\theta\cos(60^\circ + \theta)\cos(60^\circ - \theta) = \cos 3\theta \).Show Answer

32Prove that \( \cot\theta\cot(60^\circ + \theta)\cot(60^\circ - \theta) = \cot 3\theta \).Show Answer

33Prove that \( \sec(45^\circ + \frac{\theta}{2})\sec(45^\circ - \frac{\theta}{2}) = 2\sec\theta \).Show Answer

34Prove that \( \tan(\frac{\pi}{4} + \theta) - \tan(\frac{\pi}{4} - \theta) = 2\tan 2\theta \).Show Answer

35Prove that \( \sin 10^\circ\sin 50^\circ\sin 70^\circ = \frac{1}{8} \).Show Answer

36Prove that \( \cos 10^\circ\cos 50^\circ\cos 70^\circ = \frac{\sqrt{3}}{8} \).Show Answer

37Prove that \( \sin 10^\circ\sin 30^\circ\sin 50^\circ\sin 70^\circ = \frac{1}{16} \).Show Answer

38Prove that \( \sin 10^\circ\sin 50^\circ\sin 60^\circ\sin 70^\circ = \frac{\sqrt{3}}{16} \).Show Answer

39Prove that \( \cos 20^\circ\cos 30^\circ\cos 40^\circ\cos 80^\circ = \frac{\sqrt{3}}{16} \).Show Answer

40Prove that \( \cos 20^\circ\cos 40^\circ\cos 80^\circ = \frac{1}{8} \).Show Answer

41Starting from the compound angle identities, derive the formula for \( \sin C - \sin D \).Show Answer

42Express \( \sin 75^\circ\cos 15^\circ \) as a sum and find its exact value.Show Answer

43Show that \( \sin 75^\circ - \sin 15^\circ = \frac{1}{\sqrt{2}} \).Show Answer

44Without a calculator, is \( \cos 70^\circ - \cos 20^\circ \) positive or negative? Confirm it with a formula.Show Answer

45Prove that \( \frac{\sin 7A + \sin 5A}{\cos 7A + \cos 5A} = \tan 6A \).Show Answer