3.97: Cot identity and derivative of tan

0606/12/O/N/24 — Question 5 · 6 marks

(a)
Show that 1+cot2θcot2θ=sec2θ\dfrac{1+\mathrm{cot}^{2}\theta}{\mathrm{cot}^{2}\theta}=\mathrm{sec}^{2}\theta.
[1]
(b)
Write down the derivative of tan θ\mathrm{tan}\,\theta with respect to θ\theta.
[1]
(c)
Using part (a) and part (b), find the exact value of ∫0π/3(1+cot2θcot2θ−sin θ) dθ\displaystyle\int_{0}^{\pi/3}\left(\dfrac{1+\mathrm{cot}^{2}\theta}{\mathrm{cot}^{2}\theta}-\mathrm{sin}\,\theta\right)\,\mathrm{d}\theta.
[4]