a)
We first use the facts that #sinu/cosu = tanu# and #cos^2u+sin^2u =1#:
#(1+tan^2u)(1-sin^2u) = (1+sin^2u/cos^2u)(cos^2u)#
Expand:
#=(cos^2u+sin^2u)#
And done:
#=1#
b)
We need to use the cosine double angle expansions #cos(alpha-beta) = cosalphacosbeta+sinalphasinbeta# and #cos(alpha+beta)=cosalphacosbeta-sinalphasinbeta#:
using these:
#cos(alpha-beta)-cos(alpha+beta)=cosalphacosbeta+sinalphasinbeta-(cosalphacosbeta-sinalphasinbeta)#
#=2sinalphasinbeta#
c)We use the definitions of #cot# and #csc#:
#cot^2x+csc^2x=cos^2x/sin^2x+1/sin^2x#
#=(cos^2x+1)/sin^2x#
#=(1-sin^2x+1)/sin^2x#
#=(2-sin^2x)/sin^2x#
#=2csc^2x-1#
d)Now we use the definition of #sec#
#1+sec^2xsin^2x=1+sin^2x/cos^2x#
#=(cos^2x+sin^2x)/cos^2x#
#=1/cos^2x#
#=sec^2x#
