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1 Pure Mathematics

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Question 26
a.

Show that

cos⁡θ(9tan⁡θ+4tan⁡θ)≡5sin⁡θ+4sin⁡θ \cos \theta \left( 9 \tan \theta + \frac{4}{\tan \theta} \right) \equiv 5 \sin \theta + \frac{4}{\sin \theta} cosθ(9tanθ+tanθ4​)≡5sinθ+sinθ4​

for θ≠nπ2\theta \neq \frac{n\pi}{2}θ=2nπ​, where nnn is an integer.

[3]
b.

Hence solve, for 0<x<2π0 < x < 2\pi0<x<2π, the equation

cos⁡x(9tan⁡x+4tan⁡x)=12sin⁡x−2 \cos x \left( 9 \tan x + \frac{4}{\tan x} \right) = 12 \sin x - 2 cosx(9tanx+tanx4​)=12sinx−2

giving your answers to 3 significant figures.

[5]

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