Asked by Sienna
cos(arcsin(x)+arctan(x))
Answers
Answered by
Reiny
Let arcsin(x) = A
let arctan(x) = B
then cos(arcsin(x) + arctan(x))
= cos(A+B)
= cosAcosB - sinAsinB **
from arcsin(x) = A
sinA = x/1
cosA = √(1-x^2) , using Pythagoras
from arctan(x) = B
tanB = x/1 ,
then sinB = x/√(x^2 + 1)
cosB = 1/√(x^2 + 1)
back into **
cosAcosB - sinAsinB
= √(1 - x^2) (1/√(x^2 + 1) - (x/1)(x/√(x^2 + 1)
= ( √(1 - x^2) - x^2 )/(√(x^2 + 1) )
check my typing, should have written it out first.
let arctan(x) = B
then cos(arcsin(x) + arctan(x))
= cos(A+B)
= cosAcosB - sinAsinB **
from arcsin(x) = A
sinA = x/1
cosA = √(1-x^2) , using Pythagoras
from arctan(x) = B
tanB = x/1 ,
then sinB = x/√(x^2 + 1)
cosB = 1/√(x^2 + 1)
back into **
cosAcosB - sinAsinB
= √(1 - x^2) (1/√(x^2 + 1) - (x/1)(x/√(x^2 + 1)
= ( √(1 - x^2) - x^2 )/(√(x^2 + 1) )
check my typing, should have written it out first.
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