Given that $y={{\left( \tan x+\sec x \right)}^{2}}$, show that \(\cos x\frac{\mathrm{d}y

Question

Given that $y={{\left( \tan x+\sec x \right)}^{2}}$, show that $\cos x\frac{\mathrm{d}y}{\mathrm{d}x}=2y$.[2] By repeat differentiation, find the Maclaurin series for y up to and including the term in ${{x}^{3}}$.[4] Using the result obtained in (i) estimate the value of $\tan 1{}^\circ +\sec 1{}^\circ $ to $4$ decimal places.[2] By expressing $y$ in terms of sine and cosine, use the standard series in MF26 to find the series expansion for $y$ up to and including the term in ${{x}^{3}}$.[3] Comment on the results obtained from part (i) and part (iii).[1]

Accepted Answer

(i) $1+2x+2{{x}^{2}}+\frac{5}{3}{{x}^{3}}+...$(ii) $	ext{1}	ext{.0176}$ ($4$ d.p.) (iii) $1+2x+2{{x}^{2}}+\frac{5}{3}{{x}^{3}}+...$ (iv) Answers from both parts are consistent

2021 HCI P1 Q9

2021 HCI P1 Q9

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