1Which statement is correct? The choices are: a condition is needed in an identity; conditional identities are true in every case; identities are always true; conditional identities do not need to be tested.
2If \( \tan A = 1 \), what is the value of A: 0°, 30°, 45° or 60°?
3In which case is \( \sin A = \cos B \) true: A = B, \( A + B = 90^\circ \), \( A - B = 90^\circ \) or \( A + B = 180^\circ \)?
4Which of these is a conditional identity: \( \sin^2 A + \cos^2 A = 1 \); \( 1 + \tan^2 A = \sec^2 A \); if \( A + B = 90^\circ \) then \( \cot A = \tan B \); \( \sin 2A = 2\sin A\cos A \)?
5If \( A + B + C = 180^\circ \), which is true: \( \sin(A + B) = \cos C \), \( \sin(A + B) = \sin C \), \( \sin(A + B) = -\sin C \) or \( \sin(A + B) = -\cos C \)?
6The angles of a triangle are \( \alpha = 30^\circ \), \( \beta = 90^\circ \) and \( \gamma = 60^\circ \). Test \( \sin(\alpha + \beta) = \sin\gamma \).
7For the same triangle, \( \alpha = 30^\circ \), \( \beta = 90^\circ \), \( \gamma = 60^\circ \), test \( \cos(\beta + \gamma) = -\cos\alpha \).
8For \( \alpha = 30^\circ \), \( \beta = 90^\circ \), \( \gamma = 60^\circ \), test \( \tan(\alpha + \beta) = -\tan\gamma \).
9For \( \alpha = 30^\circ \), \( \beta = 90^\circ \), \( \gamma = 60^\circ \), test \( \cot(\alpha + \gamma) = -\cot\beta \).
10For \( \alpha = 30^\circ \), \( \beta = 90^\circ \), \( \gamma = 60^\circ \), test \( \sin\left(\frac{\alpha}{2} + \frac{\beta}{2}\right) = \cos\frac{\gamma}{2} \).
11For \( \alpha = 30^\circ \), \( \beta = 90^\circ \), \( \gamma = 60^\circ \), test \( \cos\left(\frac{\alpha}{2} + \frac{\beta}{2}\right) = \sin\frac{\gamma}{2} \).
12If A = 30°, B = 120° and C = 30°, test \( \tan A + \tan B + \tan C = \tan A\tan B\tan C \).
13If A = 30°, B = 120° and C = 30°, test \( \tan 2A + \tan 2B + \tan 2C = \tan 2A\tan 2B\tan 2C \).
14If A = 30°, B = 120° and C = 30°, test \( \cot A\cot B + \cot B\cot C + \cot C\cot A = 1 \).
15If A = 30°, B = 120° and C = 30°, test \( \cot 2A\cot 2B + \cot 2B\cot 2C + \cot 2C\cot 2A = 1 \).
16If A = 30°, B = 120° and C = 30°, test \( \cot 3A\cot 3B + \cot 3B\cot 3C + \cot 3C\cot 3A = 0 \).
17In triangle ABC, A = 30°, B = 90° and C = 60°. Test \( \sin 2A - \sin 2B + \sin 2C = 4\cos A\sin B\cos C \).
18In triangle ABC, A = 30°, B = 90° and C = 60°. Test \( \sin 2A - \sin 2B - \sin 2C = 4\sin A\cos B\cos C \).
19In triangle ABC, A = 30°, B = 90° and C = 60°. Test \( \cos 2A + \cos 2B + \cos 2C = -1 - 4\cos A\cos B\cos C \).
20In triangle ABC, A = 30°, B = 90° and C = 60°. Test \( \cos 2A - \cos 2B + \cos 2C = 1 - 4\sin A\cos B\sin C \).
21The angles of a triangle are \( \alpha = 45^\circ \), \( \beta = 45^\circ \) and \( \gamma = 90^\circ \). Test \( \sin^2\alpha + \sin^2\beta + \sin^2\gamma = 4\sin\alpha\sin\beta\sin\gamma \).
22For \( \alpha = 45^\circ \), \( \beta = 45^\circ \), \( \gamma = 90^\circ \), test \( \sin^2\alpha + \sin^2\beta - \sin^2\gamma = 1 - 2\cos\alpha\cos\beta\sin\gamma \).
23For \( \alpha = 45^\circ \), \( \beta = 45^\circ \), \( \gamma = 90^\circ \), test \( \cos^2\alpha - \cos^2\beta + \cos^2\gamma = 1 - 2\sin\alpha\cos\beta\sin\gamma \).
24For \( \alpha = 45^\circ \), \( \beta = 45^\circ \), \( \gamma = 90^\circ \), test \( \cos^2\alpha - \cos^2\beta - \cos^2\gamma = 2\cos\alpha\sin\beta\sin\gamma - 1 \).
25Under which condition only does \( \sin 2A + \sin 2B + \sin 2C = 4\sin A\sin B\sin C \) hold: \( A + B = 90^\circ \) and \( C = 90^\circ \); \( A + B + C = 180^\circ \); \( A + B + C = 360^\circ \); or \( A + B + C = 90^\circ \)?
26What is the difference between an identity and a conditional identity? Give one example of each.
27Show that \( \sin A + \cos B = 1 \) is not an identity.
28Without finding any values, explain why \( \tan 20^\circ = \cot 70^\circ \).
29In triangle ABC, A = 60°, B = 60° and C = 60°. Test \( \cos^2 A + \cos^2 B + \cos^2 C = 1 - 2\cos A\cos B\cos C \).