Solve Arctan Equations: Find pi Value

[mtayab1994]

Homework Statement

Solve the following equations in R.

arctan(1)=arctan(1/2)+arctan(x)
arctan(1/2)=arctan(1/3)+arctan(x)
arctan(1/3)=arctan(1/5)+arctan(x)

conclude that: pi=8arctan(1/5)+4arctan(1/7)+8arctan(1/8)

The Attempt at a Solution

For the first three i just took the tan of both sides and i got x=1/3 for the first one and x=1/7 for the second one and for the third one i got x=1/8. For the last one should i find the arctan(1/5) and the others from the first equations and plug them in or is there something else?

 

[Mark44]

Homework Statement

Solve the following equations in R.

arctan(1)=arctan(1/2)+arctan(x)
arctan(1/2)=arctan(1/3)+arctan(x)
arctan(1/3)=arctan(1/5)+arctan(x)

conclude that: pi=8arctan(1/5)+4arctan(1/7)+8arctan(1/8)

The Attempt at a Solution

For the first three i just took the tan of both sides and i got x=1/3 for the first one and x=1/7 for the second one and for the third one i got x=1/8. For the last one should i find the arctan(1/5) and the others from the first equations and plug them in or is there something else?

Substitute the values you found in the three equations:
arctan(1)=arctan(1/2)+arctan(1/3)
arctan(1/2)=arctan(1/3)+arctan(1/7)
arctan(1/3)=arctan(1/5)+arctan(1/8)

So arctan(1/3) = arctan(1) - arctan(1/2)
arctan(1/7) = arctan(1/2) - arctan(1/3)
arctan(1/8) = arctan(1/3) - arctan(1/5)

Now substitute these values into the expression 8arctan(1/5)+4arctan(1/7)+8arctan(1/8) and see if you can show that this expression simplifies to $\pi$.