(sec x + tan x)(1 - sin x) tan x = sin x

I changed (sec x + tan x) into (1/cosx + sinx/cosx), but have no idea where to go from there =(

Please help?

Also the same with the following:

(sec x + csc x) / (sin x + cos x) = 2 / (sin 2x)

Thanks

You can also convert tan(x) into sin(x)/cos(x) to get:

(1/cos(x) + sin(x)/cos(x))(1 - sin(x)) * sin(x)/cos(x)

We can then combine those expressions and terms to get:

(1 + sin(x))/cos(x) * sin(x)(1 - sin(x))/cos(x)
= sin(x)(1 + sin(x))(1 - sin(x))/cos²(x)
= sin(x)(1 - sin²(x))/cos²(x)

Recall that the Pythagorean identity is:

sin²(x) + cos²(x) = 1

Rearrange the terms:

cos²(x) = 1 - sin²(x)

So:

sin(x)(1 - sin²(x))/cos²(x)
= sin(x)cos²(x)/cos²(x)
= sin(x)
= RHS

I hope this helps!