1+sin (a) / 1+csc (a) = tan (a) / sec (a)

Below.

Step-by-step explanation:

(1 + sin a) / (1 + csc a)

= (1 + sin a) / (1 + 1 / sin a)

= (1 + sin a) / (sin a + 1) / (sin a)

= (sin a (1 + sin a)) / (1 + sin a)

= sin a.

tan a / sec a

= (sin a / cos a) / (1 / cos a)

= (sin a cos a) / cos a

= sin a

The two expressions are both equal to sin a:

Therefore

(1 + sin a) / (1 + csc a) = tan a / sec a.