Now we prove Machin's formula using the tangent addition formula:
tan(A+B)= tanA+tanB/1-tanAtanB.
If A= arctan(120/119) and B= -arctan(1/239), how do you show that
arctan(120/119)-arctan(1/239)=arctan1?
get
Mathematics
To prove that arctan(120/119) - arctan(1/239) = arctan(1), we can use the tangent addition formula:
tan(A - B) = (tan A - tan B) / (1 + tan A tan B)
Let's substitute A = arctan(120/119) and B = -arctan(1/239) into this formula:
tan(arctan(120/119) - (-arctan(1/239))) = (tan(arctan(120/119)) - tan(-arctan(1/239))) / (1 + tan(arctan(120/119)) * tan(-arctan(1/239)))
Since tan(-x) = -tan(x), we can simplify the equation:
tan(arctan(120/119) + arctan(1/239)) = (tan(arctan(120/119)) + tan(arctan(1/239))) / (1 - tan(arctan(120/119)) * tan(arctan(1/239)))
Notice that tan(arctan(x)) = x, so we can further simplify:
tan(arctan(120/119) + arctan(1/239)) = (120/119 + 1/239) / (1 - (120/119) * (1/239))
Now, simplify the right side of the equation:
(120/119 + 1/239) / (1 - (120/119) * (1/239))
= (120/119 + 1/239) / (1 - 120/119 * 1/239)
= (120/119 + 1/239) / (1 - 120/28741)
= (120 * 239 + 1) / (28741 - 120)
= (28741) / (28621)
= 1
Therefore, tan(arctan(120/119) + arctan(1/239)) = 1. By taking the inverse tangent (arctan) of both sides, we get:
arctan(120/119) + arctan(1/239) = arctan(1)
Finally, rearranging the equation, we have:
arctan(120/119) - arctan(1/239) = arctan(1)
And that concludes the proof.