IF x , y , z (x , y , z NOT EQUAL 0) ARE IN A P AND - Maths - Sequences and Series

Question

IF x , y , z (x , y , z NOT EQUAL 0) ARE IN A P AND TAN
-1 x , TAN -1 y TAN -1 z ARE ALSO
IN A P THEN PROOVE THAT x = y = z

Santosh Kumar · Accepted Answer

x, y, z are in A P 2y = x + z (1) $\tan^{- 1} x, \tan^{- 1} y, \tan^{- 1} z are in A P 2 \tan^{- 1} y = \tan^{- 1} x + \tan^{- 1} z \tan^{- 1} (\frac{2 y}{1 - y^{2}}) = \tan^{- 1} (\frac{x + z}{1 - x z}) \Rightarrow \frac{2 y}{1 - y^{2}} = \frac{x + z}{1 - x z} \Rightarrow \frac{2 y}{1 - y^{2}} = \frac{2 y}{1 - x z} [using equation (1)] \Rightarrow 1 - y^{2} = 1 - x z \Rightarrow 1 - 0 (\frac{x + z}{2})^{2} = 1 - x z$ $\Rightarrow (\frac{x + z}{2})^{2} = x z \Rightarrow (x + z)^{2} = 4 x z \Rightarrow x^{2} + 2 x z + z^{2} - 4 x z = 0 \Rightarrow x - 2 x z + z^{2} = 0 \Rightarrow (x - z)^{2} = 0 \Rightarrow x = z Now, y = \frac{x + z}{2} = \frac{2 z}{2} = z$ âˆ´ x = y = z

IF x , y , z (x , y , z NOT EQUAL 0) ARE IN A.P. AND TAN -1 x , TAN -1 y TAN -1 z ARE ALSO IN A.P. THEN PROOVE THAT x = y = z.

IF x , y , z (x , y , z NOT EQUAL 0) ARE IN A.P. AND TAN ^-1 x , TAN ^-1y TAN ^-1 z ARE ALSO IN A.P. THEN PROOVE THAT x = y = z.