Question

If x, y, z are in

a.p. and tan−1x, tan−1y and tan−1z are also in

a.p., then

Answer 1

HELLO DEAR,

YOUR QUESTIONS IS-----------------> If x, y, z are in a.p. and tan−1x, tan−1y and tan−1z are also in

a.p., then prove that:- x = y = z

GIVEN:-

x , y , z are in a.p.

so, y - x = z - y

2y = x + z----------( 1 )

similarly,

tan−1x, tan−1y and tan−1z are also in a.p.

so, 2tan^{-1}y = tan^{-1}x + tan^{-1}z

⇒tan^{-1} (2y/1 - y²) = tan^{-1} (x + z/1 - xz)

⇒(2y/1 - y²) = (x + z/1 - xz)

⇒(x + z/1 - y²) = (x + z/1 - xz)-------from( 1 )

⇒(2y/1 - y²) = (2y/1 - xz)

⇒1/1 - y² = 1/1 - xz

⇒1 - xz = 1 - y²

⇒1 - xz = 1 - {(x + z)/2}²-----from--( 1 )

⇒x² + z² + 2xz = 4xz

⇒x² + z² - 2xz = 0

⇒(x - z)² = 0

⇒x = z [put in -------( 1 ) ]

2y = x + z

⇒2y = 2x

⇒y = x

hence,  x = y = z

I HOPE ITS HELP YOU DEAR,

THANKS