Find series for ${\sec ^2}x$, in terms of ${a_1}$, ${a_3}$ and ${a_5}$, up to and including the ${x^4}$ term

by differentiating the above series for $\tan x$;

Find series for ${\sec ^2}x$, in terms of ${a_1}$, ${a_3}$ and ${a_5}$, up to and including the ${x^4}$ term

by using the relationship ${\sec ^2}x = 1 + {\tan ^2}x$.

Hence, by comparing your two series, determine the values of ${a_1}$, ${a_3}$ and ${a_5}$.

$({\sec ^2}x = ){\text{ }}{a_1} + 3{a_3}{x^2} + 5{a_5}{x^4} + \ldots$     A1

[1 mark]

${\sec ^2}x = 1 + {({a_1}x + {a_3}{x^3} + {a_5}{x^5} + \ldots )^2}$

$= 1 + a_1^2{x^2} + 2{a_1}{a_3}{x^4} + \ldots$     M1A1

Note:     Condone the presence of terms with powers greater than four.

[2 marks]

equating constant terms: ${a_1} = 1$     A1

equating ${x^2}$ terms: $3{a_3} = a_1^2 = 1 \Rightarrow {a_3} = \frac{1}{3}$     A1

equating ${x^4}$ terms: $5{a_5} = 2{a_1}{a_3} = \frac{2}{3} \Rightarrow {a_5} = \frac{2}{{15}}$     A1

[3 marks]