Prove:-
arctan(1/4)+ arctan(1/9)= (1/2)arccos(3/5)
Using the identity arctanA + arctanB = arctan((A+B)/(1-AB)),
arctan(1/4)+ arctan(1/9)
= arctan((1/4 + 1/9)/(1 - 1/36))
= arctan(13/35)
Now,
Using the identity
arctanx = (1/2)arccos((1-x^2)/(1+x^2))
=> arctan(13/35) = (1/2)arccos(1-(13/35)^2/1+(13/35)^2)
= (1/2)arccos(35^2-13^2/35^2+13^2)
= (1/2)arccos(1225-169/1225+169)
= (1/2)arccos(1056/1394)
I'm getting a slightly different answer. Perhaps one of the tutors could proofread my result
Maybe it's because the equation is false.
arctan(1/4)+arctan(1/9) = 0.3556
(1/2)arccos(3/5) = 0.4636
To prove this equation, we'll use the properties and identities of trigonometric functions. Let's break it down step by step.
1. Start with the left-hand side of the equation: arctan(1/4) + arctan(1/9).
2. Recall the identity for the sum of two tangent angles:
arctan(a) + arctan(b) = arctan((a + b) / (1 - ab))
3. Apply the identity to the left-hand side of the equation:
arctan(1/4) + arctan(1/9) = arctan((1/4 + 1/9) / (1 - (1/4)(1/9)))
4. Simplify the expression inside the arctan function:
arctan((9 + 4) / (36 - 1) = arctan(13/35)
5. Simplify further if needed, but note that arctan(13/35) cannot be expressed in terms of common trigonometric functions.
Moving on to the right-hand side of the equation:
1. Start with the right-hand side of the equation: (1/2)arccos(3/5).
2. Recall the identity for arccosine:
arccos(x) = arctan(sqrt(1 - x^2) / x)
3. Apply the identity to the right-hand side of the equation:
(1/2)arccos(3/5) = (1/2)arctan(sqrt(1 - (3/5)^2) / (3/5))
4. Simplify the expression inside the arctan function:
(1/2)arctan(sqrt(1 - 9/25) / (3/5)) = (1/2)arctan(sqrt(16/25) / (3/5))
5. Simplify further:
(1/2)arctan(4/3)
Although we have simplified both sides of the equation, it is evident that the left-hand side (arctan(1/4) + arctan(1/9)) does not equal the right-hand side [(1/2)arccos(3/5)]. Therefore, the given equation (arctan(1/4) + arctan(1/9) = (1/2)arccos(3/5)) is not true.