How do you simplify \cos(\arctan(x))?

Betsy Rhone 2021-12-15 Answered
How do you simplify cos(arctan(x))?
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Expert Answer

Melinda McCombs
Answered 2021-12-16 Author has 38 answers
Explanation:
Let simplify cos(arctan(x))
Let y=arctan(x)
x=tan(y)
x=sin(y)cos(y)
We need to have an expression for cos(y) only,
x2=sin(y)2cos(y)2
x2+1=sin(y)2+cos(y)2cos(y)2
1x2+1=cos(y)2
1x2+1=cos(y)=cos(arctan(x))
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Buck Henry
Answered 2021-12-17 Author has 33 answers
Explanation:
Let a=arctan(x)
The principal value of a(π2,π2)
Then tanα=x and
0cosα[0,1) (wrongly marked as [0,-1], in my previous answer, two years ago).
Now the given expression is
cosα=11+x2,x(π2,π2)
It is important that cosα0, for αQ1 or Q4
If the piecewise-wholesome general inverse operator
(tan1) is used
cos(tan)1x,=±11+x2
the negative sign is chosen, when xQ3
Example:
cos(arctan1)=12,arctan1=π4
cos(tan)1=cos(kπ+π4,k=0,±1,±2,±,
Q1 or Q3,x=π4,54π,
So, the value is ±12
My intention, in this approach, is to inform about nuances.
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