3.5: Sec–tan equation, identity and cot from cosec

0606/22/F/M/22 — Question 7 · 10 marks

In this question, all angles are in radians.

(a)

Solve the equation

\mathrm{sec}\,\theta = \mathrm{tan}\,\theta + 3

for

-\pi \leqslant \theta \leqslant \pi

[5]

(b)

Show that, for

0 < \phi < \dfrac{\pi}{2}

\dfrac{\mathrm{cos}\,\phi}{1 - \mathrm{sin}\,\phi} - \mathrm{tan}\,\phi = \mathrm{sec}\,\phi

[3]

(c)

Given that

\mathrm{cosec}\,x = -\dfrac{8}{17}

and that

\dfrac{\pi}{2} < x < \pi

, find the exact value of

\mathrm{cot}\,x

[2]