Can anyone help me prove these problems?

Question

1) Prove Cot(a+b)=cotacotb-1
--------------
cotb+cota

Answer 1

cot (A + B) = 1/tan(A + B) =

1 - tanA tanB
----------------- =
tan A + tan B

1 - 1/(cotAcotB) .. cotAcotB
-------------------- X ------------ =
1/cotA + 1/cotB .. cotAcotB

cotAcotB - 1
---------------- ............ QED
cotB + cotA

Answer 2

Not sure which identities you're allowed to use to prove this, but anyway...

cot(a+b) = cos(a+b) / sin(a+b)
= [cos a cos b - sin a sin b ] / [ sin a cos b + cos a sin b ]
= [cos a cos b - sin a sin b ] / [ sin a cos b + cos a sin b ] * [ 1 / (sin a sin b) ] / [ 1 / (sin a sin b) ]

= [ (cos a cos b) / (sin a sin b) - 1 ] / [ (sin a cos b) / (sin a sin b) + (cos a sin b) / (sin a sin b) ]
= [ cot a cot b - 1 ] / [ (cos b/sin b) + (cos a/sin a)]
= (cot a cot b - 1) / (cot b + cot a)