How do you simplify Tan(sec^-1(u))?

2 Answers
Hari S.
Apr 27, 2018

y/x

Explanation:

Trigonometric functions of an angle can be thought of as ratios of the sides of a triangle and its inverse functions are functions of a ratio of that triangle that return an angle.

Let there be a right triangle with vertical leg y, horizontal leg x and hypotenuse r.
Let an angle theta be defined as the angle formed from the intersection of x and r measured counter-clockwise.

sec(theta) = 1/cos(theta) = r/x
so sec^-1(u)=sec^-1(r/x)=theta
The problem now becomes tan(theta).

We know that tan(theta)=sin(theta)/cos(theta)=(y/cancel(r))/(x/cancel(r)) = y/x

Shreya
Apr 27, 2018

see below

Explanation:

Let sec^-1(u) be equal to some theta.
Thus,
sec theta=u
Squaring both sides,we get,
sec^2 theta=u^2
tan^2theta+1=u^2 [we know,tan^2theta+1=sec^2 theta]
Thus,
tan theta=+-sqrt(u^2-1)

Getting back to the original question,we have,
tan(sec^-1(u)) which on simplification gives,
tan theta which is equal to +-sqrt(u^2-1)

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