How to solve tan(x)=(cos(x)-sin(x))/(cos(x)+sin(x))?

2 Answers
Jun 9, 2018

x in {(4k+1)pi/8 | k in ZZ}.

Explanation:

In order that tanx be defined, cosx!=0.

:. Dividing the Nr. and Dr. of the R.H.S. by cosx!=0, we have,

tanx={(cosx-sinx)/cosx}/{(cosx+sinx)/cosx}.

:. tanx={cosx/cosx-sinx/cosx}/{cosx/cosx+sinx/cosx}.

:. tanx=(1-tanx)/(1+tanx)={tan(pi/4)-tanx}/{1+tan(pi/4)tanx).

:. tanx=tan(pi/4-x).................(star).

But, tantheta=tanalpha rArr theta=kpi+alpha, k in ZZ.

:. (star) rArr x=kpi+(pi/4-x), or, 2x=(4k+1)pi/4, k in ZZ.

:. x in {(4k+1)pi/8 | k in ZZ}.

Jun 9, 2018

x = pi/4 + kpi

Explanation:

tan x = (cos x - sin x)/(cos x + sin x)
Reminder of identities:
cos x - sin x = sqrt2cos (x + pi/4)
cos x + sin x = sqrt2sin (x + pi/4)
Hence,
tan x = cot (x + pi/4) = tan (pi/2 - x - pi/4) = tan (pi/4 - x)
Unit circle and property of tan function -->
x = pi/4 - x + kpi
2x = pi/4 + kpi
x = pi/8 + kpi