 In this video, we present the solution to question number seven for practice exam number two for math 1060, in which case we're asked to rewrite the expression, the trigonometric expression tangent of arc cosine of x as an equivalent expression involving only the variable x. So when it comes to these type of exercises, we have an inverse trigonometric function inside of a trigonometric function, your best bet is to draw yourself a right triangle and then use that diagram to analyze what's going on here. So basically what we're going to do is we're going to associate with this inverse trigonometric function, the angle theta, or more specifically we're going to say theta is equal to cosine inverse of x. Then by the inverse function property, this is equivalent to saying that cosine of theta is equal to x. We'd like to think of trig functions giving us a ratio so we'll think this is x over one. And so with associated angle theta this right triangle has as its adjacent side x and as its hypotenuse one. Well, we have that this side squared plus this side squared equals this side squared using the Pythagorean equation, we can solve for the remaining side right here to give you the square root of one minus x squared. Basically you have this other side here y, excuse me, y, you have that x squared plus y squared equals one squared. If you solve for y, you'll get exactly this expression right here. Again, just using the Pythagorean equation. So with this triangle we can label any of the trig ratios. Which trig ratio do we want? We want tangent of theta, right? So tangent of course is opposite over adjacent. So let's use the opposite side. We found out it to be the square root of one minus x squared over the adjacent side which is x. So we look amongst the answers to that and we see that choice D would be the correct answer.