Question

prove it:

tan`¹x+tan`¹y+tan`¹z=tan`¹ x+y+z-xyz/1-xy-yz-zx​

Answer 1

Answer:

Given, tan^-1x + tan^-1y + tan^-1z = π

tan^-1x + tan^-1y = π - tan^-1z

we know, tan^-1P + tan^-1Q = tan^-1(P + Q)/(1 - PQ) , where PQ < 1

so, tan^-1x + tan^-1y = tan^-1(x + y)(1 - xy)

so, tan^-1(x + y)/(1 - xy) = π - tan^-1z

(x + y)/(1 - xy) = tan(π - tan^-z)

we know, tan(π - ∅) = - tan∅

so, (x + y)/(1 - xy) =- tan(tan^-1(z))

=> (x + y)/(1 - xy) = -z

=> (x + y) = (1 - xy)(-z) = -z + xyz

=> x + y + z = xyz [ hence proved

Read more on Brainly.in - https://brainly.in/question/5596429#readmore

Answer 2

Answer:

tan`¹x=alfa

tan`¹=beta

tan`¹=gamma

tan alfa=x, tan beta=y, tan gamma=z

:.now,

tan(alfa+beta+gamma)

= tan alfa+tan beta+tan gamma-tan alfa tan beta tan gamma

=x+y+z-xyz/1-xy-yz-zx

:.alfa + beta+ gamma=tan`¹ x+y+z-xyz/1-xy-yz+zx

or, tan`¹x+tan`¹y+tan`¹z=tan`¹ x+y+z-xyz/1-xy-yz-zx

(prove)

(hope it's helpful☺)