Question

2tan(x)(sin(x)+cos(x))=sec(x)(1+sin2x-cos2x) how to prove this?

verify, prove this

Answer 1

Since the right hand side is more elaborate, we will modify the right hand side to match that of the left hand side...see below

Explanation:

`2tanx(sinx+cosx)=secx(1+sin2x-cos2x)`

Double angle for sin and cos:
`sin2x= 2sinxcosx`
`cos2x= 1-2sin^2x`

`2tanx(sinx+cosx)=secx(1+(2sinxcosx)-(1-2sin^2x))=`

`2tanx(sinx+cosx)=secx(cancel(1)+2sinxcosxcancel(-1)+2sin^2x)`

Apply the reciprocal identity:
`secx= 1/cosx`

`2tanx(sinx+cosx)=1/cosx(2sinxcosx+2sin^2x)`

Distribute:
`2tanx(sinx+cosx)=(2sinxcosx)/(cosx)+(2sin^2x)/cosx`

`2tanx(sinx+cosx)=(2sinxcosx+2sin^2x)/cosx`

GCF:
`2tanx(sinx+cosx)=(2sinx(cosx+sinx))/cosx`

Quotient Indentity:
`tanx=sinx/cosx`

Therefore:
`2tanx(sinx+cosx)=2tanx(sinx+cosx)`