Question. Simplify tan(cos−1 x).
Solution. First write y = cos−1 x. Then cos y = x and consider the triangle
below.
y
x
The trick is to let the hypotenuse of this triangle be equal to 1, and use the
Pythagorean theorem. So if the hypotenuse is 1, then
a2 + b2 = (hypotenuse)2 =⇒ x2 + b2 = 1.
Solving for b2 means b2 = 1 − x2 , so b =
√
1 − x2 . Putting this information
back into the triangle gives
1
√
1 − x2
y
x
So since tangent of an angle is the opposite side over the adjacent side, we have
√
tan(cos
−1
x) = tan(y) =
1
1 − x2
.
x
To see this another way, write tan y =
tan(cos−1 x) =
sin y
cos y .
Then
sin(cos−1 x)
sin(cos−1 x)
=
.
−1
cos(cos x)
x
Unfortunately, now you have to use the same trick as before to get that
√
sin(cos
−1
x) =
p
1 − x2
= 1 − x2 .
1
In class, I mentioned that there may be a way to do this using trigonometric
identities, but that way seems to cumbersome (or impossible?) so this is the
best approach.
2