Trig Identities to prove equations

Question

a.) Using the identity for tan(x+y), prove tan^(-1)x + tan^(-1)y = tan^(-1)[(x+y) / (1-xy)]

where -π/2 < tan^(-1)x + tan^(-1)y < π/2

b.) Use part a to show that tan^-1 (1/2) + tan^(-1)(1/3) = π/4

c.) Use the first four terms of the Maclaurin series of tan^(−1) x and part (b) to approximate the value of π

Answer 1

a) x = tan(X) and y = tan(Y)

tan(X+Y) = (tanX + tanY)/(1-tanXtanY)

Take the inverse tan of both sides and substitute x and y into right-hand side

X+Y = tan^-1((x+y)/(1-xy))

b) Just plug in. x=1/2 and y=1/3 in to the right-hand-side expression and you get tan^-1(1) = pi/4

c) Google or look up Maclaurin series for tan^-1x. Plug in the value of the first four terms tan^-1(x) and tan^-1(y) with x = 1/2, y = 1/3, add them and multiiply by 4 to get an approximation for pi.

pi = 4(tan^-1(1/2) + tan^-1(1/3))

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