Let be defined for .

MATHEMATICS · Multi-part · A Level · Mathematics · Differentiation · Differentiation

Problem

Let

g(x)=\tan x

be defined for

x\in\left(-\tfrac{\pi}{2},\tfrac{\pi}{2}\right)

. (1) Rewrite

g(x)

exclusively in terms of

\sin x

and

\cos x

. No other trigonometric functions should appear. (2) Differentiate the expression you obtained in part (a) to obtain

g'(x)

. Show all algebraic steps. (3) Simplify

g'(x)

so that it is expressed as a single elementary trigonometric function of

x

. Justify every simplification step clearly.