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]