((1+sin x)/cos x) + (cos x/(1+sin x)) = 2sec x

Lefthand Side

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

Combine the two fractions into one fraction only,

= [(1 +sin x)(1 +sin x) +(cos x)(cos x)] / [(cos x)(1 +sin x)]

= [(1 +2sin x +sin^2 x) +(cos^2 x)] / [(cos x)(1 +sin x)]

= [1 +2sin x +sin^2 x +cos^2 x] / [(cos x)(1 +sin x)]

Since sin^2 x +cos^2 x = 1, then,

= [1 +2sin x +1] / [(cos x)(1 +sin x)]

= [2 +2sin x] / [(cos x)(1 +sin x)]

= [2(1 +sin x] / [(cos x)(1 +sin x)]

= [2] / [(cos x)]

= 2sec x

= Righthand Side

Verified.