$\def\D{\displaystyle}$

1 (Edexcel, Further Pure Math, 2013 jan, Paper 2, No 2)

Using the identities $\D

\sin (A + B) = \sin A \cos B + \cos A \sin B \\

\cos (A + B) = \cos A \cos B – \sin A \sin B \\

\tan A=\D\frac{\sin A}{\cos A} $

(a) show that $\D \tan(A+B)=\frac{\tan A+\tan B}{1-\tan A\tan B}$ (3)

(b) Hence show that

(i) $\D \tan 105^{\circ}=\frac{1+\sqrt{3}}{1-\sqrt{3}}$

(ii) $\D \tan 15^{\circ}=\frac{\sqrt{3}-1}{1+\sqrt{3}}$ (4)

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