Question

If cos x = 4/5, 0° x 90°, find the value
of (1 + tan x)/(1 - tan x)

Answer 1

cos x = 4 / 5

sin x = ±√ ( 1 - sin^2 x )

The angles which lie between 0° and 90° are lie in the first quadrant.

In the first quadrant sin x is positive so:

sin x = √ ( 1 - sin^2 x )

sin x = √ ( 1 - sin^2 x ) = sin x = √ [ 1 - ( 4 / 5 ) ^ 2 ] =

√ [ 1 - 16 / 25 ] = √ [ 25 / 25 - 16 / 25 ] =

√ [ 9 / 25 ] = √ 9 / √ 25 = 3 / 5

sin x = 3 / 5


Now:

tan x = sin x / cos x

tan x = ( 3 / 5 ) / ( 4 / 5 ) = 3 * 5 / 4 * 5 = 3 / 4

tan x = 3 / 4


( 1 + tan x ) / ( 1 - tan x ) =

( 1 + 3 / 4 ) / ( 1 - 3 / 4 ) =

( 4 / 4 + 3 / 4 ) / ( 4 / 4 - 3 / 4 ) =

( 7 / 4 ) / ( 1 / 4 ) = 7 * 4 / 1 * 4 = 7 / 1 = 7

Answer 2

In QI, if cosx = 4/5, sinx = 3/5

(1+tanx)/(1-tanx) = (cosx+sinx)/(cosx-sinx)
= (4/5 + 3/5)/(4/5 - 3/5)
= (7/5)/(1/5)
= 7