Compound, double-angle and reciprocal identities

Compound and reciprocal identities

Choose an identity and preserve its domain

1. Compound Sine uses matching signs; cosine uses the opposite cross-term sign
2. Double Use the sine or cosine double-angle identity \(\displaystyle \sin 2A=2\sin A\cos A,\qquad\cos 2A=1-2\sin^2 A=2\cos^2 A-1\)
3. Reciprocal Keep the stated denominator restrictions \(\displaystyle \begin{aligned}1+\tan^2 A&=\sec^2 A&&\text{for }\cos A\ne0\\1+\cot^2 A&=\csc^2 A&&\text{for }\sin A\ne0\end{aligned}\)
4. Proof Transform one side using justified identities and record any nonzero denominator

Worked example

Find the exact tangent using the angle difference shown

1. \(\displaystyle \frac{\sqrt3-1}{\sqrt3+1}=\frac{(\sqrt3-1)^2}{3-1}=2-\sqrt3\)Both input tangents and the nonzero denominator are defined. Rationalise without rounding

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