1.6: Trigonometric identities and equations

4PM1/2/June/2022 — Question 7 · 11 marks

(i) (a) Using a formula from page 2, show that
tan 2θ=2tan θ1−tan2θ.\mathrm{tan}\,2\theta = \frac{2\mathrm{tan}\,\theta}{1-\mathrm{tan}^2\theta}.
(2)
Given that tan 2α=1\displaystyle \mathrm{tan}\,2\alpha=1,
(b) show that
tan α=a±b,\mathrm{tan}\,\alpha=a\pm\sqrt{b},
where a\displaystyle a and b\displaystyle b are integers whose values need to be found.
(3)
(ii) (a) Using formulae from page 2, show that
cos (x−30)∘=sin (x+30)∘\mathrm{cos}\,(x-30)^\circ=\mathrm{sin}\,(x+30)^\circ
can be written as tan x∘=1\displaystyle \mathrm{tan}\,x^\circ=1.
(4)
(b) Hence, or otherwise, solve
cos (2y−30)∘=sin (2y+30)∘\mathrm{cos}\,(2y-30)^\circ=\mathrm{sin}\,(2y+30)^\circ
for −90<y≤90\displaystyle -90<y\leq90.
(2)