How do you sove for all values of x: sin2x-tanx= tanx cos2x?

1 Answer
Øko
Jan 24, 2018

It is true for all values of #x#

Explanation:

It is true for all values of #x#

It is the easiest to start from the right side:

We will use:

  • #cos^2+sin^2=1#

  • #cos(2x)=cos^2(x)-sin^2(x)#

  • #sin(2x)=2cos(x)sin(x)#

Working out

#RHS=tan(x)cos(2x)#

#=tan(x)(cos^2(x)-sin^2(x))#

#=tan(x)cos^2(x)-tan(x)sin^2(x)#

#=tan(x)cos^2(x)-tan(x)(1-cos^2(x))#

#=tan(x)cos^2(x)-tan(x)+tancos^2(x)#

#=2tan(x)cos^2(x)-tan(x)#

#=2cos(x)sin(x)-tan(x)#

#=sin(2x)-tan(x)#

#=LHS#

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