Proving Trig Identities?

Question

Establish the identity.

Show that tan^-1 (x) + cot^-1 (x) = (pi/2)

Answer 1

tan^-1 (x) + cot^-1 (x) = (pi/2)

Let p=tan^-1 (x) then x=tanp

Let q=cot^-1 (x) then x=cot q

p+q=?

x=tanp=cot q

p=q=45=pi/4

p+q=pi/4+pi/4=2pi/4=pi/2

Answer 2

Let tan^-1(x) = y
=> x = tan y
=> x = cot(pi/2 - y)
=>cot^-1(x) = pi/2 - y
=>cot^-1(x) + y = pi/2
=>cot^-1(x) + tan^-1(x) = pi/2

Answer 3

if tan^-1 (x) =theta, then cot^-1 (x)=pi/2 -theta
so, tan^-1 (x) + cot^-1 (x)=theta + pi/2 - theta =pi/2