Trigonometry verify equation is identity

Question

Sinx+cosx/sinx-cosx= 1+2sinx×cosx/2sin^2x-1

I keep getting stuck on these after a few lines

Answer 1

Recall that sin²x + cos²x = 1. So, cos²x = 1 - sin²x.

Multiply the numerator and denominator of (sinx + cosx) / (sinx - cosx) by sinx + cosx to get:

[sin²x + 2sinxcosx + cos²x] / [sin²x - cos²x]

= [1 + 2sinxcosx] / [sin²x - (1 - sin²x)] = [1 + 2sinxcosx] / [2sin²x - 1]

Answer 2

(sinx +cosx)/(sinx- cosx) = (1 +2sinxcosx)/(2sinx^2(x) -1)

left side

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

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

= (sin2x +1)/-cos2x

right side

(sin2x +1)/-cos2x

it's an identity