Simplify (1+csc x) / cos x+cot x

(1 + cscx) / cosx+cotx = (1+cscx) / (cosx+cosx/sinx)

= (1+cscx) / cosx (1+1/sinx)

= (1+cscx) / cosx (1+cscx)

= 1/cosx

= secx

(1+cscx) / (cos x + cot x)

As cosec x is the inverse of sinx

csc x = 1/sinx,

and cotx is the ratio of cosx and sinx

cotx = cosx/sinx

(1+cscx) / (cos x + cot x)

(1 + 1/sinx) / (cosx + cosx/sinx)

BY taking LCM:

(sinx + 1) / sinx * [ (sinxcosx + cosx) / sinx]

(sinx + 1) / (sinxcosx + cosx)

Taking cosx as common:

= (sinx + 1) / cosx * (sinx + 1)

= 1 / cosx is the answer