Specialist Mathematics Β· Unit 2 Β· Trigonometry and functions Β· Trigonometric identities

Prove and apply the identities for products of sines and cosines expressed as sums and differences. sin(𝐴) sin(𝐡) = 1 2 (cos(𝐴 βˆ’ 𝐡) βˆ’ cos(𝐴 + 𝐡)) cos(𝐴) cos(𝐡) = 1 2 (cos(𝐴 βˆ’ 𝐡) + cos(𝐴 + 𝐡)) sin(𝐴) cos(𝐡) = 1 2 (sin(𝐴 + 𝐡) + sin(𝐴 βˆ’ 𝐡)) cos(𝐴) sin(𝐡) = 1 2 (sin(𝐴 + 𝐡) βˆ’ sin(𝐴 βˆ’ 𝐡))

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Prove and apply the angle sum, difference and double-angle identities for sines and cosines. sin(𝐴 + 𝐡) = sin(𝐴) cos(𝐡) + cos(𝐴) sin(𝐡) sin(𝐴 βˆ’ 𝐡) = sin(𝐴) cos(𝐡) βˆ’ cos(𝐴) sin(𝐡) cos(𝐴 + 𝐡) = cos(𝐴) cos(𝐡) βˆ’ sin(𝐴) sin(𝐡) cos(𝐴 βˆ’ 𝐡) = cos(𝐴) cos(𝐡) + sin(𝐴) sin(𝐡) sin(2𝐴) = 2 sin(𝐴) cos(𝐴) cos(2𝐴) = cos2(𝐴) βˆ’ sin2(𝐴) = 1 βˆ’ 2 sin2(𝐴) = 2 cos2(𝐴) βˆ’ 1
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Prove and apply the Pythagorean identities. sin2(𝐴) + cos2(𝐴) = 1 tan2(𝐴) + 1 = sec2(𝐴) cot2(𝐴) + 1 = cosec2(𝐴)
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