How do you prove: 2 tan 2 x = (sin x + cos x )/ (cos x − sin x) − (cos x − sin x)/ (cos x + sin x) ?

1 Answer
Feb 28, 2018

See below.

Explanation:

Identities:

1) #color(red)(bb(sin(2x)=2sinxcosx)#

2) #color(red)bb(cos(2x)=2cos^2(x)-1)#

3) #color(red)bb(tan(2x)=sin(2x)/cos(2x))#

4) #color(red)bb(sin^2x+cos^x=1)#

#RHS#

Add the two fractions:

#(sinx+cosx)/(cosx-sinx)-(cosx-sinx)/(cosx+sinx)#

#((sinx+cosx)(sinx+cosx)-(cosx-sinx)(cosx-sinx))/((cosx-sinx)(cosx+sinx))#

Expand and simplify denominator:

#(sin^2x+2sinxcosx+cos^2x-cos^2x+2sinxcosx-sin^2x)/(cos^2x-sin^2x)#

Simplify numerator:

#->(4sinxcosx)/(cos^2x-sin^2x)#

Using identity 1

#->(2sin(2x))/(cos^2x-sin^2x)#

Using identity 4

#->(2sin(2x))/(cos^2x-(1-cos^2x)#

#->(2sin(2x))/(2cos^2x-1)#

Using identity 2

#->(2sin(2x))/(cos(2x))#

Using identity 3

#2tan(2x)#

#RHS-=LHS#