Write an expression that is equivalent to sec2tan1x yet that

Write an expression that is equivalent to sec(2tan^-1x), yet that doesn\'t use trigonometric functions. Assume that the variable represents a postive value.

Solution

sec(2tan^-1(x)

Let 2tan^-1x = y

tan^-1(x) = y/2

x = tan(y/2)

cos(y/2) = 1/sqrt(1 +x^2)

cosy = 2cos^2(y/2) -1 = 2(1/(1 +x^2) -1

= ( 2 - 1 - x^2)/( 1 +x^2)

= ( 1 - x^2)/(1 +x^2)

secy = (1 +x^2)/(1 - x^2)

So,sec(2tan^-1(x) = secy

= (1 +x^2)/(1 - x^2)