Target Mathematics

1.33: Cosine–tangent to quadratic in sine

0606/13/O/N/20 — Question 11 · 7 marks

(a)
Given that 2cos⁡x=3tan⁡x2\cos x = 3\tan x, show that 2sin⁡2x+3sin⁡x−2=02\sin^{2}x + 3\sin x - 2 = 0.
[3]
(b)
Hence solve 2cos⁡(2α+π4)=3tan⁡(2α+π4)2\cos\left(2\alpha + \tfrac{\pi}{4}\right) = 3\tan\left(2\alpha + \tfrac{\pi}{4}\right) for 0<α<π0 < \alpha < \pi radians, giving your answers in terms of π\pi.
[4]