1Which of these is equal to sin A: \( \cos^2\frac{A}{2} + \sin^2\frac{A}{2} \), \( \cos^2\frac{A}{2} - \sin^2\frac{A}{2} \), \( 1 - 2\cos^2\frac{A}{2} \) or \( 2\sin\frac{A}{2}\cos\frac{A}{2} \)?
2Which of these is equal to cos θ: \( 4\sin\frac{\theta}{3} - 3\sin^3\frac{\theta}{3} \), \( 4\cos^3\frac{\theta}{3} - 3\cos\frac{\theta}{3} \), \( 3\cos\frac{\theta}{3} - 4\cos^3\frac{\theta}{3} \) or \( 3\sin\frac{\theta}{3} - 4\sin^4\frac{\theta}{3} \)?
3If A = 60°, what is the value of \( \sin\frac{A}{2} \): 0, 1, \( \frac{1}{2} \) or \( \frac{\sqrt{3}}{2} \)?
4Which of these is a submultiple angle of 60°: 15°, 25°, 35° or 45°?
5Which of these is equal to tan A: \( \frac{2\tan\frac{A}{2}}{1 + \tan^2\frac{A}{2}} \), \( \frac{1 - \tan^2\frac{A}{2}}{1 + \tan^2\frac{A}{2}} \), \( \frac{2\tan\frac{A}{2}}{1 - \tan^2\frac{A}{2}} \) or \( \frac{1 + \tan^2\frac{A}{2}}{1 - \tan^2\frac{A}{2}} \)?
6Define a submultiple angle with examples.
7Express sin θ in terms of \( \tan\frac{\theta}{2} \).
8Express cos θ in terms of \( \cos\frac{\theta}{2} \).
9Express cos θ in terms of \( \tan\frac{\theta}{2} \).
10Express sin θ in terms of \( \sin\frac{\theta}{3} \).
11If \( \sin\frac{A}{2} = \frac{4}{5} \), find sin A, cos A and tan A.
12If \( \tan\frac{A}{2} = \frac{3}{4} \), find sin A, cos A and tan A.
13If \( \cos\frac{A}{2} = \frac{5}{13} \), find sin A, cos A and tan A.
14If \( \sin\frac{\theta}{3} = \frac{1}{2} \), find sin θ, cos θ and tan θ.
15If \( \cos\frac{\theta}{3} = \frac{4}{5} \), find sin θ, cos θ and tan θ.
16If \( \tan\frac{\theta}{3} = \frac{1}{2} \), find tan θ.
17If \( \cos A = \frac{7}{25} \), find \( \sin\frac{A}{2} \).
18If \( \cos A = \frac{13}{36} \), find \( \cos\frac{A}{2} \).
19If \( \tan A = \frac{4}{3} \), find \( \tan\frac{A}{2} \).
20If \( \cos 30^\circ = \frac{\sqrt{3}}{2} \), prove that \( \sin 15^\circ = \frac{\sqrt{3} - 1}{2\sqrt{2}} \).
21If \( \cos 30^\circ = \frac{\sqrt{3}}{2} \), prove that \( \cos 15^\circ = \frac{\sqrt{3} + 1}{2\sqrt{2}} \).
22If \( \cos 30^\circ = \frac{\sqrt{3}}{2} \), prove that \( \tan 15^\circ = 2 - \sqrt{3} \).
23If \( \cos 30^\circ = \frac{\sqrt{3}}{2} \), prove that \( \cot 15^\circ = 2 + \sqrt{3} \).
24If \( \cos 45^\circ = \frac{1}{\sqrt{2}} \), prove that \( \cos 22\frac{1}{2}^\circ = \frac{1}{2}\sqrt{2 + \sqrt{2}} \).
25If \( \cos 45^\circ = \frac{1}{\sqrt{2}} \), prove that \( \sin 22\frac{1}{2}^\circ = \frac{1}{2}\sqrt{2 - \sqrt{2}} \).
26If \( \cos 45^\circ = \frac{1}{\sqrt{2}} \), prove that \( \tan 22\frac{1}{2}^\circ = \sqrt{3 - 2\sqrt{2}} \).
27If \( \cos 45^\circ = \frac{1}{\sqrt{2}} \), prove that \( \cot 22\frac{1}{2}^\circ = \sqrt{2} + 1 \).
28If \( \cos\frac{\theta}{2} = \frac{1}{2}\left(a + \frac{1}{a}\right) \), prove that \( \cos\theta = \frac{1}{2}\left(a^2 + \frac{1}{a^2}\right) \).
29If \( \sin\frac{A}{2} = \frac{1}{2}\left(a + \frac{1}{a}\right) \), prove that \( \cos A = -\frac{1}{2}\left(a^2 + \frac{1}{a^2}\right) \).
30If \( \sin\frac{\beta}{3} = \frac{1}{2}\left(a + \frac{1}{a}\right) \), prove that \( \sin\beta = -\frac{1}{2}\left(a^3 + \frac{1}{a^3}\right) \).
31If \( \cos\frac{\beta}{3} = \frac{1}{2}\left(a + \frac{1}{a}\right) \), prove that \( \cos\beta = \frac{1}{2}\left(a^3 + \frac{1}{a^3}\right) \).
32Prove that \( \frac{1 + \cos A}{\sin A} = \cot\frac{A}{2} \).
33Prove that \( \frac{\sin A}{1 + \cos A} = \tan\frac{A}{2} \).
34Prove that \( \frac{1 - \cos A}{\sin A} = \tan\frac{A}{2} \).
35Prove that \( \frac{\sin A}{1 - \cos A} = \cot\frac{A}{2} \).
36Prove that \( \frac{1 + \sin A}{\cos A} = \frac{\cos\frac{A}{2} + \sin\frac{A}{2}}{\cos\frac{A}{2} - \sin\frac{A}{2}} \).
37Prove that \( \cos A = \frac{1 - \tan^2\frac{A}{2}}{1 + \tan^2\frac{A}{2}} \).
38Prove that \( \frac{1 - \tan\frac{\theta}{2}}{1 + \tan\frac{\theta}{2}} = \frac{1 - \sin\theta}{\cos\theta} \).
39Prove that \( \frac{\cos\alpha}{1 + \sin\alpha} = \frac{1 - \tan\frac{\alpha}{2}}{1 + \tan\frac{\alpha}{2}} \).
40Prove that \( \frac{1 + \sin\theta - \cos\theta}{1 + \sin\theta + \cos\theta} = \tan\frac{\theta}{2} \).
41Prove that \( \frac{\sin\theta - \cos\frac{\theta}{2}}{1 - \sin\frac{\theta}{2} - \cos\theta} = \cot\frac{\theta}{2} \).
42Prove that \( \sin A + \tan A = \frac{4\tan\frac{A}{2}}{1 - \tan^4\frac{A}{2}} \).
43Prove that \( \tan\frac{A}{2} + \cot\frac{A}{2} = 2\,\text{cosec}\,A \).
44Prove that \( \tan\left(45^\circ + \frac{A}{2}\right) = \frac{\cos A}{1 - \sin A} \).
45Prove that \( 2\sin^2\left(45^\circ - \frac{A}{2}\right) = 1 - \sin A \).
46Prove that \( 2\cos^2\left(45^\circ - \frac{A}{2}\right) = 1 + \sin A \).
47Prove that \( \cos^2\left(45^\circ - \frac{\theta}{2}\right) - \sin^2\left(45^\circ - \frac{\theta}{2}\right) = \sin\theta \).
48Prove that \( \frac{1 - \tan^2(45^\circ - A)}{1 + \tan^2(45^\circ - A)} = \sin 2A \).
49Prove that \( \frac{1 - \tan^2\left(45^\circ - \frac{A}{2}\right)}{1 + \tan^2\left(45^\circ - \frac{A}{2}\right)} = \sin A \).
50Prove that \( \tan\left(45^\circ + \frac{A}{2}\right) = \sec A + \tan A = \sqrt{\frac{1 + \sin A}{1 - \sin A}} \).
51Prove that \( \frac{\sin 2A}{1 + \cos 2A}\times\frac{\cos A}{1 + \cos A} = \tan\frac{A}{2} \).
52Prove that \( \left(1 + \cos\frac{\pi}{8}\right)\left(1 + \cos\frac{3\pi}{8}\right)\left(1 + \cos\frac{5\pi}{8}\right)\left(1 + \cos\frac{7\pi}{8}\right) = \frac{1}{8} \).
53If \( \tan\frac{A}{2} = \frac{1}{2} \), find sin A, cos A and tan A.
54Is \( \sin\frac{A}{2} \) the same as \( \frac{1}{2}\sin A \)? Check with A = 60°.