Question

Prove That ,


$\frac{ \sin(5x) - 2 \sin(3x) + \sin(x) }{ \cos(5x) - \cos(x) } = \tan(x)$
​

Answer 1

Step-by-step explanation:

(sin5x-2sin3x+sinx)/(cos5x-cosx)

=[{2sin(5x+x)/2cos(5x-x)/2}-2sin3x]/{2sin(5x+x)/2sin(x-5x)/2}

=(2sin3xcos2x-2sin3x)/{2sin3xsin(-2x)}

={2sin3x(cos2x-1)}/{-2sin3xsin2x}

=-(cos2x-1)/sin2x

=(1-cos2x)/sin2x

=2sin²x/2sinxcosx

=sinx/cosx

=tanx (Proved)

Answer 2

↪Solution↩

sin5x-2sin3x+sinx)/(cos5x-cosx)

=> [{2sin(5x+x)/2cos(5x-x)/2}-2sin3x]/{2sin(5x+x)/2sin(x-5x)/2}

=> (2sin3xcos2x-2sin3x)/{2sin3xsin(-2x)}

=> {2sin3x(cos2x-1)}/{-2sin3xsin2x}

=> -(cos2x-1)/sin2x

=> (1-cos2x)/sin2x

=> 2sin²x/2sinxcosx

=> sinx/cosx

=> tanx (Proved)