# Prove that (cos x + sin x) / (cos x - sin x) = tan [x+(pi/4)]

justaguide | College Teacher | (Level 2) Distinguished Educator

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The identity to be proven is (cos x + sin x) / (cos x - sin x) = tan [x+(pi/4)]

Start from the right hand side

tan [x+(pi/4)]

expand tan (x + pi/4)

=> (tan x + tan pi/4) / (1 - tan x* tan pi/4)

use tan pi/4 = 1

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

substitute tan x = sin x/ cos x

= [(sin x / cos x) + 1] / [1 - (sin x / cos x)]

= [(sin x / cos x) + (cos x / cos x)] / [(cos x / cos x) - (sin x / cos x)]

= [(sin x + cos x) / cos x] / [(cos x - sin x) / cos x)]

= [(sin x + cos x) / cos x] * [cos x / (cos x - sin x)]

= (sin x + cos x) / (cos x - sin x)

This is the left hand side.

This proves (cos x + sin x) / (cos x - sin x) = tan [x+(pi/4)]

lochana2500 | Student, Undergraduate | (Level 1) Valedictorian

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R:H:S ≡ tan(x + π/4 )

= (tanx + tan π/4) / (1-tanx.tan π/4)

= (tanx + 1) / (1-tanx)

= (sinx/cosx + 1) / (1 - sinx/cosx)

= [(sinx + cosx)/cosx] / [(cosx - sinx)/cosx]

= [(sinx + cosx)/cosx] * [cosx/(cosx - sinx)]

= (cosx + sinx)/(cosx- sinx)

= L:H:S