Verify the identity 11 sin x 11 sin x 2 sec2 xSolution1

Verify the identity 1/1 + sin x + 1/1 - sin x = 2 sec^2 x.

1 / ( 1+ sin x ) + 1 / ( 1 - sin x ) = 2 sec^2 x

manipulating the left hand side

taking lcd as ( 1 + sin x )( 1- sin x )

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

since denominators are equal we can now add the numerators

( 1 - sinx + 1 + sin x ) / ( 1 + sin x )( 1- sin x )

( 1 + sin x )( 1- sin x ) = 1- sin^2 x

so we get

2 / ( 1 - sin^2 x )

from the identity

cos^2 x = 1 - sin^2 x

2 / cos^2 x = 2 sec^2 x

which is our right side

proved !