Solve this?

If sin^(-1)(x-x^2/x+x^3/4.....)+cos^(-1)(x^2-x^4/2+x^6/4.....)=pi/2 for 0<x<sqrt2 then x equals

a. 1

b. 1/2

c. 0

d.-1

1 Answer
Mar 20, 2018

a. 1

Explanation:

sin^-1 theta+cos^-1theta=pi/2
You have:
sin^-1(x-x^2/2+x^3/4-...)+cos^-1(x^2-x^4/2+x^6/4-...)=pi/2

Thus, we can say,
(x-x^2/2+x^3/4-...)=(x^2-x^4/2+x^6/4-...)
[because sin^-1 theta+cos^-1theta=pi/2; so theta is the common or same angle]

From the equation, we understand:
x=x^2, x^2=x^4,x^3=x^6, and so on.
These can be possible only when (x=1) or when (x=0).

color(blue) (0< x < sqrt2.
Thus, as x>0, the only possible value of x is 1.