for the first question:

cot x [(tan y + tan x)/(cot x + cot y)]

remembering the rules:

cot a = cos a/sin a

tan a = sin a/cos a

hence the equation can now be written as

=(cosx/sinx)[ {(sin y/cos y) + (sin x/cos x)} / {(cos y/sin y) + (cos x/sin x)}]

=(cosx/sinx)[{(cos x.sin y+cos y.sin x)(sin y.sin x)} / {(cos y.cos.x)/(cos y.sin x + cos x.sin y)}

= (cos x/sin x)[{sin y.sin x}/{cos y.cos x}]

= sin y/cos y

= tan y