3.101: Derivative of tan x − x and integral

0606/21/O/N/24 — Question 2 · 4 marks

(a)
Given that y=tan x−xy = \mathrm{tan}\,x - x, find dydx\dfrac{\mathrm{d}y}{\mathrm{d}x}. Write your answer in terms of tan x\mathrm{tan}\,x.
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(b)
Hence find ∫0π/4tan2x dx\displaystyle\int_{0}^{\pi/4}\mathrm{tan}^{2} x\,\mathrm{d}x. Give your answer in exact form.
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