SOLUTION: Use a right triangle to write the expression as an algebraic expression. Assume that x is positive and in the domain of the given inverse trigometric function. cos(tan-1 x)

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Question 324168: Use a right triangle to write the expression as an algebraic expression. Assume that x is positive and in the domain of the given inverse trigometric function.
cos(tan-1 x)
Thank you.
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
If you draw a triangle, you'll find that the two legs are 'x' and '1' which will make the hypotenuse to be sqrt%28x%5E2%2B1%29. The opposite leg is 'x' and the adjacent leg is '1' which means that tan%28theta%29=x%2F1=x or simply that tan%28theta%29=x. Take the arctangent of both sides to get theta=arctan%28x%29 (note: this is a simplified view of the arctangent function)


So essentially, cos%28arctan%28x%29%29=cos%28theta%29. So because the adjacent side is 1 unit long and the hypotenuse is sqrt%28x%5E2%2B1%29, this means that cos%28theta%29=1%2Fsqrt%28x%5E2%2B1%29


So


Again, a drawing really helps you see why this is true.