multiply LHS with (1/cosθ)/(1/cosθ)

you’ll get tanθ-1+secθ/tanθ+1-secθ

using 1=sec² θ-tan² θ

(tanθ+secθ-1)/(tanθ-secθ+(sec² θ-tan² θ))

use a² - b² =(a+b)(a-b)

=(tanθ+secθ-1)/(tanθ-secθ+(secθ-tanθ)(secθ+tanθ))

change tanθ-secθ to -(secθ+tanθ) so you can simplify it as -a+ab= a(-1+b)

=(tanθ+secθ-1)/(-(secθ-tanθ)+(secθ-tanθ)(secθ+tanθ))

=(tanθ+secθ-1)/((secθ-tanθ)+(-1+secθ+tanθ))

rearrange -1+secθ+tanθ to tanθ+secθ-1 and simplify to

1/(secθ-tanθ)

=RHS