Mathematics

Problem

Two curves are given by

y_1=\sin x + x

and

y_2 = x\tan x

for

0<x<\dfrac{\pi}{2}

. a. (i) Find the expressions for the slopes of the tangents to

y_1

and

y_2

at any point

x

in the stated interval. b. (ii) Show that the tangents to the two curves are parallel when

\sin x = x\sec^2 x -1

and hence find the common

x

–coordinate of such points, correct to two decimal places. c. (iii) Determine the acute angle between the two curves at

x=\dfrac{\pi}{4}