Can you please verify this trig identity

tan x + ((cos x) / (1+sin x)) = sec x

sinx/cosx + cosx/(1+sinx) =
(sinx(1+sinx) + cosx^2)/(cosx(1+sinx)) =
(sinx + sinx^2 + cosx^2)/(cosx(1+sinx)) =
(sinx + 1)/(cosx(sinx+1)) =
1/cosx =
secx

(cos(x))/(1+sin(x))+tan(x) = sec(x)

Put tan(x)+(cos(x))/(sin(x)+1) over the common denominator sin(x)+1: tan(x)+(cos(x))/(sin(x)+1) = (cos(x)+(1+sin(x)) tan(x))/(1+sin(x)):
(cos(x)+tan(x) (1+sin(x)))/(1+sin(x)) = ^?sec(x)

Multiply both sides by sin(x)+1:
cos(x)+(1+sin(x)) tan(x) = ^?sec(x) (1+sin(x))


Write secant as 1/cosine and tangent as sine/cosine:
(sin(x)+1) (sin(x))/(cos(x))+cos(x) = ^?(sin(x)+1) 1/(cos(x))

Put cos(x)+(sin(x) (sin(x)+1))/(cos(x)) over the common denominator cos(x): cos(x)+(sin(x) (sin(x)+1))/(cos(x)) = (cos(x)^2+sin(x) (1+sin(x)))/(cos(x)):
(cos(x)^2+sin(x) (1+sin(x)))/(cos(x)) = ^?(1+sin(x))/(cos(x))

Multiply both sides by cos(x):
cos(x)^2+sin(x) (1+sin(x)) = ^?1+sin(x)

cos(x)^2 = 1-sin(x)^2:
1-sin(x)^2+sin(x) (1+sin(x)) = ^?1+sin(x)

Expand sin(x) (1+sin(x)).
sin(x) (1+sin(x)) = sin(x)^2+sin(x):
1-sin(x)^2+sin(x)+sin(x)^2 = ^?1+sin(x)

Evaluate 1-sin(x)^2+sin(x)+sin(x)^2.
1-sin(x)^2+sin(x)+sin(x)^2 = sin(x)+1:
1+sin(x) = ^?1+sin(x)