Solve { 2+2sin2x } /{ 2(1+sinx)(1-sinx) } =sec^2x+tanx ?

2 Answers
May 9, 2018

x = k \pi quad integer k

Explanation:

Solve { 2+2sin2x } /{ 2(1+sinx)(1-sinx) } =sec^2x+tanx

0 = { 2+2sin2x } /{ 2(1+sinx)(1-sinx) } - sec^2x - tanx

= { 2+2 (2 sin x cos x) } /{ 2(1-sin ^2 x) } - 1/cos^2x - sin x/cos x

= {1 + 2 sinx cos x}/{cos ^2 x} - 1/cos^2 x - {sin x cos x}/cos^2 x

= {sin x cos x}/{cos ^2 x} = tan x

tan x = 0

x = k \pi quad integer k

May 9, 2018

x=kpi,kinZZ

Explanation:

We have,

(2+2sin2x)/(2(1+sinx)(1-sinx))=sec^2x+tanx

=>(2(1+sin2x))/(2(1-sin^2x))=sec^2x+tanx

=>(1+sin2x)/cos^2x=sec^2x+tanx

=>1+sin2x=sec^2xcos^2x+tanxcos^2x

=>1+sin2x=1+sinx/cosx xxcos^2x

=>sin2x=sinxcosx

=>2sin2x=2sinxcosx

=>2sin2x=sin2x

=>2sin2x-sin2x=0

=>color(red)(sin2x=0...to(A)

=>2x=kpi,kinZZ

=>x=(kpi)/2, kinZZ
But, for this x ,sinx=1=>1-sinx=0
So,
(2+2sin2x)/(2(1+sinx)(1-sinx))=(2+0)/(2(1+1)(0))=2/0to undefined

Thus,
x!=(kpi)/2, kinZZ
Hence, there is no solution.!!
Again from (A)

sin2x=0=>2sinxcosx=0=>sinxcosx=0
=>sinx=0 or cosx=0,where,tanx and secx is defined.
i.e. cosx!=0=>sinx=0=>color(violet)(x=kpi,kinZZ
There is contradiction in result when we take sin2x=0.