SOLUTION: If arctan(x)=arccos(y), show that {{{y =1/sqrt(1+x^2)}}}.

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Question 1153180: If arctan(x)=arccos(y), show that
y+=1%2Fsqrt%281%2Bx%5E2%29.
Answer by Edwin McCravy(20064) About Me  (Show Source):
You can put this solution on YOUR website!
If arctan(x)=arccos(y), show that
y+=1%2Fsqrt%281%2Bx%5E2%29
Let the angle that both sides equal be θ: 

theta=arctan%28x%2F1%29=arccos%28y%2F1%29

So

matrix%281%2C3%2Ctan%28theta%29=x%2F1%2Cand%2Ccos%28theta%29=y%2F1%29

and since 

matrix%281%2C3%2Ctangent=opposite%2Fadjacent%2C+and%2C+cosine=adjacent%2Fhypotenuse%29,

we can draw two similar right triangles with angle θ, one
for each side of the equation.



We use the Pythagorean theorem to find the missing sides:



Because they are similar right triangles,





matrix%281%2C3%2Cy%2C%0D%0A%22%22=%22%22%2C%0D%0A1%5E%22%22%2Fsqrt%281%2Bx%5E2%29%29

Edwin