Target Mathematics

1.72: Trigonometric identities and exact trigonometric values

4PM1/2/January/2017 — Question 4 · 10 marks

tan⁡(A+B)=tan⁡A+tan⁡B1−tan⁡Atan⁡B\tan(A+B)=\frac{\tan A+\tan B}{1-\tan A\tan B}
(a) (i) Write down an expression for tan⁡(2x)\tan(2x) in terms of tan⁡x\tan x.
(ii) Hence show that tan⁡(3x)=3tan⁡x−tan⁡3x1−3tan⁡2x\displaystyle\tan(3x)=\frac{3\tan x-\tan^3x}{1-3\tan^2x}. (6)
Given that α\alpha is the acute angle such that cos⁡α=13\cos\alpha=\frac13
(b) find the exact value of tan⁡α\tan\alpha. (2)
(c) Hence use the identity in part (a) to find the exact value of tan⁡(3α)\tan(3\alpha).
Give your answer in the form a2b\displaystyle\frac{a\sqrt2}{b} where aa and bb are integers. (2)