Question 1
Which identity correctly simplifies the expression shown in the diagram?
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Specialist Mathematics Β· Unit 2 Β· Trigonometry and functions Β· Trigonometric identities
Prove and apply the angle sum, difference and double-angle identities for sines and cosines. sin(π΄ + π΅) = sin(π΄) cos(π΅) + cos(π΄) sin(π΅) sin(π΄ β π΅) = sin(π΄) cos(π΅) β cos(π΄) sin(π΅) cos(π΄ + π΅) = cos(π΄) cos(π΅) β sin(π΄) sin(π΅) cos(π΄ β π΅) = cos(π΄) cos(π΅) + sin(π΄) sin(π΅) sin(2π΄) = 2 sin(π΄) cos(π΄) cos(2π΄) = cos2(π΄) β sin2(π΄) = 1 β 2 sin2(π΄) = 2 cos2(π΄) β 1
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Question 2
Prove that \[ \cos(A - B) = \cos(A)\cos(B) + \sin(A)\sin(B) \] using the angle sum identity for cosine and the properties of even and odd functions.
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Question 3
Given that \(\sin(\theta) = \frac{3}{5}\) where \(0 < \theta < \frac{\pi}{2}\) and \(\cos(\phi) = \frac{5}{13}\) where \(0 < \phi < \frac{\pi}{2}\), what is the exact value of \(\sin(\theta + \phi)\)?
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Prove and apply multi-angle trigonometric identities up to angles of 4π₯ using the identities listed above, e.g. cos(4π₯) = 8 cos4(π₯) β 8 cos2(π₯) + 1 and cosec(2π₯) β cot(2π₯) = tan(π₯). Specialist Mathematics 2025 v1.4
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Prove and apply the identities for products of sines and cosines expressed as sums and differences. sin(π΄) sin(π΅) = 1 2 (cos(π΄ β π΅) β cos(π΄ + π΅)) cos(π΄) cos(π΅) = 1 2 (cos(π΄ β π΅) + cos(π΄ + π΅)) sin(π΄) cos(π΅) = 1 2 (sin(π΄ + π΅) + sin(π΄ β π΅)) cos(π΄) sin(π΅) = 1 2 (sin(π΄ + π΅) β sin(π΄ β π΅))